# Does bilinear models on vectors mean dot or outer product?

If I have 2 vectors $$x$$ and $$y$$ where $$x \in \mathcal{R}^{m}$$ and $$y \in \mathcal{R}^{n}$$.
Does bilinear model mean?
$$f(x,y) = x^TWy$$ where $$W \in \mathcal{R}^{m*n}$$
which result in a scalar
or
$$f(x,y) = W(x⊗y^T)$$ where ⊗ is the outer product and $$W \in \mathcal{R}^{m*n}$$.
which result in a matrix

I checked 2 papers, the first one Low-rank Bilinear Pooling in page 2 in equation 1 their bilinear model produce a scalar
while in Compact Bilinear Pooling in section 3.1 they said "Bilinear models take the outer product of two vectors"

## 1 Answer

"Bilinear" is simply an adjective that you can apply to any function of two vectors to indicate that it is linear in each argument. So the linear map

$$(x,y) \rightarrow x^TWy$$

and the tensor product

$$(x,y) \rightarrow xy^T$$

can both be described as bilinear, even though their codomains are different.

• What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does). – floyd Jan 7 at 20:20