diff --git a/PackageInfo.g b/PackageInfo.g
index cca34fa..a83ba77 100644
--- a/PackageInfo.g
+++ b/PackageInfo.g
@@ -10,7 +10,7 @@ SetPackageInfo( rec(
PackageName := "MachineLearningForCAP",
Subtitle := "Exploring categorical machine learning in CAP",
-Version := "2024.07-17",
+Version := "2024.07-20",
Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ),
License := "GPL-2.0-or-later",
diff --git a/README.md b/README.md
index 6277692..c1f0490 100644
--- a/README.md
+++ b/README.md
@@ -42,3 +42,502 @@ To obtain current versions of all dependencies, `git clone` (or `git pull` to up
[code-img]: https://img.shields.io/badge/-View%20code-blue?logo=github
[code-url]: https://github.com/homalg-project/MachineLearningForCAP#top
+
+### Introduction
+Let $A$ and $B$ be two sets and $\Theta$ be a set of parameters. A parametrized map is a map of the form $f:\Theta\times A \to B$ where $A$ is the domain of the input variables, $\Theta$ the domain of parameters and $B$ the codomain. For each fixed $\theta \in \Theta$, we get the map $f_\theta:A \to B,~~a \mapsto f_\theta(a):=f(\theta,a)$.
+
+In machine learning, both predictions maps and loss maps can be seen as parametrized maps, any they play distinct but complementary roles in the model training process. Let us break down how each of these fits into the concept of parametrized maps.
+
+Let $\mathcal{D}$ be the **training set** containing $N$ _labeled examples_: $\mathcal{D}=$ { $(x^{[i]},y^{[i]})|i=1,2,\dots,N$ } $\subset X\times Y$.
+- $x^{[i]}$ is the input-vector (feature vector) and it belongs to the input-space $X$.
+- $y^{[i]}$ is the label-vector of $x^{[i]}$ and it belongs to the output-space $Y$.
+
+A **prediction map** can be defined as a parametrized map $f:\Theta \times X \to Y$. For a given and parameters-vector $\theta\in\Theta$ and an input-vector $x\in X$, the prediction is $\hat{y}=f_\theta(x)\in Y$.
+
+For example let $\mathcal{D}=$ { $(1,2.9), (2,5.1), (3,7.05)$ } $\subset \mathbb{R}^1\times \mathbb{R}^1 \simeq \mathbb{R}^2$. For a linear regression model, the prediction might be defined as follows:
+
+$$f:\mathbb{R}^2\times \mathbb{R}^1 \to \mathbb{R}^1;~~~(\theta,x) \mapsto \theta_1 x + \theta_2$$
+
+where $\theta=(\theta_1,\theta_2)\in \mathbb{R}^2$ is a parameter-vector. For instance, if $\theta=(2,1)$, then the prediction associated to the input $x=2$ is $\hat{y}=2\cdot 2 + 1=5$.
+
+A **loss map** can be defined as a parametrized map $\ell:\Theta \times X \times Y \to \mathbb{R}$. It quantifies the discrepancy between the predicted output $\hat{y}=f_\theta(x) \in Y$ and the true output $y\in Y$. Given a training set, the **total loss** map is defined by
+
+$$\mathcal{L}:\Theta \to \mathbb{R};~~~\theta \mapsto \frac{1}{N}\sum_{i=1}^{N}\ell(\theta,x^{[i]},y^{[i]})$$
+
+where $N=|\mathcal{D}|$ (the number of labeled examples in the training set).
+
+For a linear regression model, the loss map can be defined as follows:
+
+$$\ell:\mathbb{R}^2\times \mathbb{R}^1 \times \mathbb{R}^1 \to \mathbb{R}, (\theta,x,y) \mapsto (\hat{y} - y)^2=(\theta_1 x + \theta_2 - y)^2$$
+
+where $\theta=(\theta_1,\theta_2)\in \mathbb{R}^2$ is a parameter-vector. For instance, if $\theta=(2,1)$, then the total loss on the above training set $\mathcal{D}$ is $\frac{(2\cdot 1 + 1-2.9)^2+(2\cdot 2 + 1-5.1)^2+(2\cdot 3 + 1-7.05)^2}{3}=0.0074$.
+
+Given a training set $\mathcal{D}\subset X\times Y$ and a loss map $\ell:\Theta \times X \times Y \to \mathbb{R}$, we would like to "learn" a parameter-vector $\theta\in\Theta$ such that the total loss $\mathcal{L}(\theta)$ is as small as possible. We illustrate the learning procedure via the next two examples.
+
+## Example 1: Neural Network to Perform Linear-Regression
+
+In this example, we consider a training dataset consisting of the three points $\mathcal{D}=$ { $( 1, 2.9 ), ( 2, 5.1 ), ( 3, 7.05 )$ } $\subset\mathbb{R}^2$.
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