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What is the difference between Fully Connected layers and Bilinear layers in deep learning?

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I quote the answers from What is a bilinear tensor layer (in contrast to a standard linear neural network layer) or how can I imagine it?.

A bilinear function is a function of two inputs $x$ and $y$ that is linear in each input separately. Simple bilinear functions on vectors are the dot product or the element-wise product.

Let $M$ be a matrix. The function $f(x,y)=x^TMy=\sum_iM_{ij}x_iy_j$ is bilinear in $x$ and $y$. In fact, any scalar bilinear function on two vectors takes this form. Note that a bilinear function is a linear combination of $x_iy_j$ whereas a linear function such as $g(x,y)=Ax+By$ can only have $x_i$ or $y_i$. For neural nets, that means a bilinear function allows for richer interactions between inputs.

Now what if you want a bilinear function that outputs a vector? Well, you simply define a matrix $M_k$ for each coordinate of the output and you end up with a stack of matrices. That stack of matrices is called a tensor (3-mode tensor to be exact). You can imagine the bilinear tensor product with two vectors as $x^⊀M_ky$ computed on each β€œslice” of the tensor.

Bilinear Models consists of two feature extractors whose outputs are multiplied using an outer product at each location of the image and pooled to obtain an image descriptor. 1

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Its advantage is that it can model pairwise feature interactions in a translationally invariant manner, which is particularly useful for fine-grained categorization. It also allows end-to-end training using image labels only, and achieves state-of-the-art performance on fine-grained classification.

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    $\begingroup$ Great summary. Just one point -- for the Bilinear CNN model we directly took an outer product of x and y, which would mean an identity matrix for M. Since x and y are the result of learned projections anyway, one can assume anything that a matrix M would encode can be learned end-to-end in the features. $\endgroup$
    – AruniRC
    Commented Jan 31, 2019 at 18:33
  • $\begingroup$ I am stuck in the formula 𝑓(π‘₯,𝑦)=π‘₯𝑇𝑀𝑦=βˆ‘π‘–π‘€π‘–π‘—π‘₯𝑖𝑦𝑗. Can someone explain it? How do we multiply for instance $x^{T}A$. The dimensions don't seems to match. I am not sure how to do this matching actually since x is 2d and M is 3d. $\endgroup$ Commented Feb 10, 2021 at 6:50
  • $\begingroup$ For each image, they are done separately. You can search for vectorization. $\endgroup$ Commented Feb 10, 2021 at 7:01
  • $\begingroup$ @EduardoReis it is not simple matrix multiplication. It is done by einstein summation aka einsum. $\endgroup$
    – Ritwik
    Commented Jan 6, 2023 at 10:58
  • $\begingroup$ @GreenFalcon does bilinear functions works better than simple concatenation (of $x$ and $y$) and a linear layer? If yes, then why? $\endgroup$
    – Ritwik
    Commented Jan 6, 2023 at 11:00

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