A Derivation of the 3-entity-type model

The 3-entity-type model consists of two relationship matrices, a m× n matrix X and a n× r matrix Y . In the spirit of generalized linear models we define the low rank representations of the relationship matrices to be X ≈ f1(UV T ) and Y ≈ f2(V Z T ) where f1 : R m×n → R and f2 : R n×r → R are the prediction links, and U ∈ R, V ∈ R, and Z ∈ R are the parameters of the model for k ∈ Z. We further define parameter transformations G : R → R, H : R → R, I : R → R, to model prior knowledge about our parameters, e.g., regularizers. We additionally required the convex conjugate,