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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"

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"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.

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  • $\begingroup$ 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). $\endgroup$ – floyd Jan 7 at 20:20

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