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 :).
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 $\beta$.