# Index

$g()$ is any ***Non-Linear Function***

## Basic Architecture

### Multiplicative Modules

With these modules we can modify our ***traditional*** ways of ***neural networks***
and implement ***switch-like*** functions

#### Professor's one

Basically here we want a ***way to modify `weights` with `inputs`***.

Here $\vec{z}$ and $\vec{x}$ are both `inputs`

$$
\begin{aligned}
    \vec{y} &= \sum_{w_{i,j}x{j}} \\
    \vec{w} &= \sum_{k} u_{i,j,k} z_{k} \rightarrow \\
    \rightarrow \vec{y} &= \sum_{j,k} u_{i,j,k} z_{k} x_{j}
\end{aligned}
$$

As we can see here, $z_{k}$ modifies, along $u$, $x_{j}$.

#### Quadratic Layer

This layer expands data by applying the **quadratic formula**

$$
\begin{aligned}
    \vec{v} &= [a_1, a_2, a_3] \\

    quad\_layer(\vec{v}) &= [ a_1 \cdot a_1, a_1 \cdot a_2, a_1 \cdot a_3, ... , a_3 \cdot a_3 ]
\end{aligned}
$$

#### Product Unit[^product-unit]

$$
o_k =  \sum_{j}^{m} v_{k,j} \cdot \left( \prod_{i=1}^{n} x_{i}^{w_{j,i}}\right) + v_{k,0}
$$

#### Sigma-Pi Unit[^simga-pi][^simga-pi-2]

This *layer* is basically a product of `input` terms **times**
a `weight`, intead of a `matrix multiplication` of a `linear-layer`.

Moreover, this is ***not necessarily*** `fully-connected`

$$
o_k = g\left( \sum_{q \in conjunct} w_{q} \prod_{k=1}^{N} z_{q,k} \right)
$$

### Attention Modules

They define a way for our `model` to get what's ***more important***

#### Softmax

We use this function to output the ***importance*** of a certain
value over all the others.

$$
\begin{aligned}
    \sigma(\vec{x})_{j} &= \frac{e^{x_{j}}}{\sum_{k} e^{x_{k}}} \;\; \forall k \in {0, ..., N} \\

    \sigma(\vec{x})_{j} &\in [0, 1] \;\; \forall j \in {0, ..., N}
\end{aligned}
$$

## Mixture of Experts[^mixture-of-experts]

What happens if we have more `models` and we want to take their output?

Basically we have a set of `weights` over our `outputs` before the `output-layer`.
Both the **experts** and the **gating-function** need to be `trained`.

> [!TIP]
>
> Since we are talking about `weights` and `importance`, probably here it is better to use an [attention-model](#attention-modules)

## Parameter Transformation

It is basically when the `wheights` are the `output` of a ***function***

Since they are controlled by some other `parameters`, then we need to ***learn***
those instead

### Weights Sharing

Here we ***copy*** our weights over more ***basic components***.
Since we have ***more than one value*** for our `original weights`, then we need to ***sum*** those.

> [!TIP]
>
> This is used to find ***motifs*** on an `input`

<!-- Footnotes -->

[^simga-pi]: [University of Pretoria | sigma-pi | pg. 2](https://repository.up.ac.za/bitstream/handle/2263/29715/03chapter3.pdf?sequence=4#:~:text=A%20pi%2Dsigma%20network%20\(PSN,of%20sums%20of%20input%20components.)
[^simga-pi-2]:[]
[^product-unit]: doi: 10.13053/CyS-20-2-2218
[^mixture-of-experts]: [Wikipedia | 1st April 2025](https://en.wikipedia.org/wiki/Mixture_of_experts)