# 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"

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