Based on the CCG category (S\NP)/NP of a transitive verb, this is a third-order tensor, S ⊗ N ⊗ N. The relative pronoun has a CCG type of (NP\NP)/(S\NP) or (NP\NP)/(S/NP), resulting in a fourthorder tensor, N ⊗ N ⊗ S ⊗ N. In order to make the learning of a relative pronoun tensor tractable, we make use of the method introduced by Paperno, Pham, and  and Polajnar, Fagarasan, and Clark (2014) in order to reduce the semantic type of the transitive verb to S ⊗ N. This method models a transitive verb as two second-order tensors (matrices), one governing the interaction of the verb with its subject and the other with its object, and has been shown to reproduce sentence semantics without loss of accuracy.
this "decoupling" approximation for the transitive verb allows the relative pronoun to be modeled as a third-order tensor, N ⊗ N ⊗ S. we adopt the approach of Paperno, Pham, and Baroni (2014).