This paper extends supervised embedding models by combining them multiplicatively, 

i.e. $f'(x,y) = G(x,y) f(x,y). $

It considers two types of model, dot product in the *embedding* space and kernel density in the *embedding* space, where the kernel in the embedding space is restricted to 

$k((x,y),(x','y)) = k(x-x')k(y-y'). $

It proposes an iterative algorithm which alternates $f$ and $G$ parameter updates.