Currently, I am going through Geometry of the Restricted Boltzmann Machine by Cueto et al.
In section 2, the authors defines $\psi (v, h)$ as follows.
$$ \psi (v, h) = e^{(h^\top W v + b^\top v + c^\top h)} $$
Here, $v$ is the state of the visible layers, $h$ is the state of the hidden layer, and $W$ is the connectivity matrix of a restricted Boltzmann machine.
Then the authors convert this exponential function into a squarefree monomial. To do that they defines the following functions.
$$ \gamma_i = e^{c_i}\\ w_{ij} = e^{W_{ij}}\\ \beta_j = e^{b_j} $$
They the monomial is:
$$ \psi (v, h) = \Pi^k_{i = 1} \gamma^{h_i}_i \cdot \Pi^k_{i=1} \Pi^n_{j=1} w^{h_i v_j}_{ij} \cdot \Pi^n_{j=1} \beta^{v_j}_j $$
I don't understand how to get to the monomial from the exponential function. This is how I tried.
$$ \psi (v, h) = e^{(h^\top W v + b^\top v + c^\top h)}\\ = e^{h^\top W v} e^{b^\top v} e^{c^\top h}\\ = e^{h^\top W v} e^{(b^\top)^v} e^{(c^\top)^h} $$
What do I do now?
It is only a matter of exponent manipulations. Start with:
$$ \begin{align} e^{b^T v} &= e^{\sum_{j=1}^n b_j v_j} \\ &= e^{b_1 v_1} \cdot \dots \cdot e^{b_n v_n} \\ &= (e^{b_1})^{v_1} \cdot \dots \cdot (e^{b_n})^{v_n} \\ &= \prod_{j=1}^n \beta_j^{v_j} \end{align} $$
The derivation for the other terms is similar.