Bilinear and K-Mapping Basis in Linear Algebra I am reading a machine learning paper which has some mathematical terminologies that are proving a little hard for me to understand. I am going to write the lines from the papers here. 

The policy is synthesized with a bilinear mapping 
$\qquad \qquad \pi(a|s)=exp(\phi^t A(e_t, e_k) \theta)$
The bilinear mapping given by the matrix $A$ is parameterized as the
  linear combination of $K$ mapping $\Theta_k$
$\qquad \qquad A(e_t, e_k)=\sum^K_{k=1}\alpha_k(e_t, e_k)\Theta_k$
where the coefficient function $\alpha_k(\cdot,\cdot)$ are parameterized by one hidden layer
  MLP

Questions


*

*What does bilinear mapping mean here?

*What does $K$ mapping mean here?

*Assuming if $A(e_t, e_k)$ is a $4\times4$, is $\Theta_k$ one single matrix or there are $K$ different $4 \times 4$ matrices?

*What is the purpose $\Theta_k$?

*Since $\alpha_k(\cdot,\cdot)$ is a coefficient function represented by a MLP, are there $K$ different MLPs as well?

 A: 
  
*
  
*What does bilinear mapping mean here?
  

It looks like $\phi$ and $\theta$ are vectors and $A(e_t, e_k)$ is a matrix. So, the bilinear map is referring to $\phi^T A(e_t, e_k) \theta$ (which produces a real value). You can think of this as a function:
$$f(\phi, \theta) = \phi^T A(e_t, e_k) \theta$$
$f$ is a bilinear map from whatever vector space(s) $\phi$ and $\theta$ live in to the real numbers.


  
*What does $K$ mapping mean here?
  

It looks like this is a possible typo, and "mapping" is meant to be "mappings". As above, we can think of matrix $A(e_t, e_k)$ as defining a mapping. The second equation in your quoted text says that this matrix is given by a linear combination of $K$ matrices $\{\Theta_1, \dots, \Theta_K\}$ (and we could think of each $\Phi_k$ as defining a mapping in a similar sense).


  
*Assuming if $A(e_t, e_k)$ is a $4\times4$, is $\Theta_k$ one single matrix or there are $K$ different $4 \times 4$ matrices?
  

$\Theta_k$ appears in a summation where $k$ is the index. So, there are $K$ matrices: $\{\Theta_1, \dots, \Theta_K\}$


  
*What is the purpose $\Theta_k$?
  

The matrix $A(e_t, e_k)$ is formed as a linear combination of the matrices $\{\Theta_1, \dots, \Theta_k\}$. You could think of these as basis elements that are combined with different strengths to form the final matrix.


  
*Since $\alpha_k(\cdot,\cdot)$ is a coefficient function represented by a MLP, are there $K$ different MLPs as well?
  

That sounds plausible, but hard to say for sure based on the short excerpt you provided; check the rest of the paper.
A: I can offer an answer to your first question.
In a vector space, a linear map $f\colon V \rightarrow W$ is a function between the underlying sets of vectors such that $$f(x + y) = f(x) + f(y)$$ and $$f(cx)= cf(x)$$ where $x, y$ are vectors and $c$ is a scalar.
The principle behind a bilinear map is that you want linearity in two arguments. Consider the function between vector spaces $g\colon U\times V \rightarrow W$. Being linear in its two arguments means that fixing $u \in U$ gives a linear map $h_{0}\colon V \rightarrow W$ which sends $v\mapsto g((u,v))$. Similarly, fixing $v \in V$ gives a linear map $h_{1}\colon U \rightarrow W$ which sends $u\mapsto g((u,v))$. In short, you can express $g$ as two linear maps $h_{0}$ and $h_{1}$.
By the way, this leads to the concept of a tensor product in which one constructs some object $U \otimes V$ that leads to a new function which is now a (usual) linear map $U \otimes V\rightarrow W$.
