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
$f(x,y) = W(x⊗y^T)$ 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"


"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.

  • $\begingroup$ 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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