A key challenge in information theoretic feature selection is to estimate mutual information expressions that capture three desirable terms—the relevancy of a feature with the output, the redundancy and the complementarity between groups of features. The challenge becomes more pronounced in multi-target problems, where the output space is multi-dimensional. Our work presents an algorithm that captures these three desirable terms and is suitable for the well-known multi-target prediction settings of multi-label/dimensional classification and multivariate regression. We achieve this by combining two ideas—deriving low-order information theoretic approximations for the input space and using quantization algorithms for deriving low-dimensional approximations of the output space. Under the above framework we derive a novel criterion, Group-JMI-Rand, which captures various high-order target interactions. In an extensive experimental study we showed that our suggested criterion achieves competing performance against various other information theoretic feature selection criteria suggested in the literature.