However, modeling relations as mappings naturally requires more parameters -a general linear map between d-dimension vectors is represented by a matrix of d 2 parameters -which are less likely to be shared, impeding transfers of facts between similar relations.
it is desired to reduce dimensionality of relations; furthermore, the existence of a composition of two relations (assumed to be modeled by matrices M 1 , M 2 ) matching a third (M 3 ) also justifies dimension reduction, because it implies a compositional constraint M 1 • M 2 ≈ M 3 that can be satisfied only by a lower dimension sub-manifold in the parameter space 1 .