In many artificial intelligence, machine learning and computer vision tasks, the weighted sum model is used to value objects and define an order over them. In this paper, we consider two decision criteria defined as the (Euclidean and more generally Mahalanobis-like) distance to a reference point and investigate how they relate to the weighted sum model. In particular, we show that the distance-based representations can be seen as a relaxation of the representation induced by the weighted sum and we provide a characterization of the latter model with the former models in the case of strict orders. To illustrate our point, we consider the context of relative visual attributes. Nonetheless, our results also apply to other domains. More specifically, we present how these reference-point-based representations can be learned from pairwise comparisons and how they can be exploited for classification. Our experimental results show that those two criteria yield a more precise representation of the relative ordering for some attributes and that combining the best representations for each attribute improves recognition performance.