What is a bilinear form? I'm a CS master student and I'm reading a paper that mentions the term "bilinear form". Actually the paper mentions "bilinear regression model". But I think in order to understand what a "bilinear regression model" is, I need to understand what does "bilinear form" mean. I checked the wiki page on "bilinear form" but couldn't understand.


*

*Can you please explain to me (in simpler ways) the idea behind 'bilinear form'?

*If you know what 'bilinear regression model' is, I would also be very thankful for an explanation as well :).
 A: First lets cover linear regression. Using a single parameter $\alpha$ we move in a line between two points $\mathbf{x}_1$ and $\mathbf{x}_2$ with the equation
$$ \mathbf{x} = (1-\alpha)\, \mathbf{x}_1 + \alpha \, \mathbf{x}_2 $$
Here each point $\mathbf{x}$ is an $n$ dimensional vector.
This is a linear form because only the first power of $\alpha$ is used.
Now consider the case where the two control points $\mathbf{x}_1$ and $\mathbf{x}_2$ aren't fixed, but the result of linear regression themselves using a parameter $\beta$ two pairs of control points $\mathbf{x}_3$, $\mathbf{x}_4$ and $\mathbf{x}_5$, $\mathbf{x}_6$.
$$\begin{aligned} 
  \mathbf{x}_1 & = (1-\beta) \mathbf{x}_3 + \beta \mathbf{x}_4 \\
  \mathbf{x}_2 & = (1-\beta) \mathbf{x}_5 + \beta \mathbf{x}_6
\end{aligned}$$
This is bilinear form, because the location of $\mathbf{x}$ depends on two linear parameters $\alpha$ and $\beta$.
$$ \begin{aligned} \mathbf{x} & = (1-\alpha) \left( (1-\beta) \mathbf{x}_3 + \beta \mathbf{x}_4 \right) + \alpha\, \left( (1-\beta) \mathbf{x}_5 + \beta \mathbf{x}_6 \right) \\
 & =  \mathbf{A} + \mathbf{B} \alpha + \mathbf{C} \beta + \mathbf{D} \alpha\beta \\
& & \mathbf{A} = \mathbf{x}_3 \\
  & & \mathbf{B} = \mathbf{x}_5- \mathbf{x}_3 \\
 & & \mathbf{C} = \mathbf{x}_4 - \mathbf{x}_3 \\
 & & \mathbf{D} = \mathbf{x}_3+\mathbf{x}_6-\mathbf{x}_4-\mathbf{x}_5
\end{aligned}$$
I can explain the above graphically better:

All the points along the $ \overline{\mathbf{x}_3 \mathbf{x}_4}$ line have $\alpha=0$ and $\beta = 0 \ldots 1$. Similarly, the points along the $ \overline{\mathbf{x}_5 \mathbf{x}_5}$ line have $\alpha=1$ and $\beta = 0 \ldots 1$. The point $\mathbf{x}$ lies on a line with the same $\beta$ value and $\alpha=0 \ldots 1$.
It is said that there is a correspondence between the cartesian coordinates of the points and the regression parameters
$$\begin{cases}
  \mbox{point} & \mbox{parameters }(\alpha,\beta) \\
  \mathbf{x}_3 & (0,0) \\
  \mathbf{x}_4 & (1,0) \\
  \mathbf{x}_5 & (0,1) \\
  \mathbf{x}_6 & (1,1) \\
  \mathbf{x}_1 & (0,\beta) \\
  \mathbf{x}_2 & (1,\beta) \\
  \mathbf{x} & (\alpha,\beta) 
\end{cases} $$
The point $\mathbf{x}$ is bilinear mapping of the plane with coordinates $\alpha$ and $\beta$. 
A: The paper says it clearly, the bilinear model is of the form $f(x_i,x_j;V)=x_i^TVx_j$.
If you consider a linear model of a single vector variable, the most general form for a function taking scalar values can be expressed as a dot product with a constant vector $f(x_i;v)=x_i^Tv$.
The bilinear form is the generalization to two vector variables. It is linear in both variables, hence its name.
$$(a\,x+b\,y)^TVz=a\,x^TVz+b\,y^TVz$$ and
$$x^TV(a\,y+b\,z)=a\,x^TVy+b\,x^TVz.$$
The generalizations to more than two variables require tensors.
