This reduces the class of models to: However, this still leaves the particular form of the function f unspecified.
now, if we assume that p lies in the same space as u and v, avoiding the issues of dimensionality associated with tensor products, and that f is a linear function, for simplicity, of the cartesian product of u and v, then we generate a class of additive models: where A and B are matrices which determine the contributions made by u and v to the product p. if we assume that f is a linear function of the tensor product of u and v, then we obtain multiplicative models: where C is a tensor of rank 3, which projects the tensor product of u and v onto the space of p. Further constraints can be introduced to reduce the free parameters in these models.