added images

wiki/MachineLearning/Support-Vector-Machines.md

@@ -15,23 +15,32 @@ Repository: https://github.com/pharo-ai/Support-Vector-Machines
 Support Vector Machines (abbreviated as SVM) are supervised learning algorithms that can be used for classification and regression problems. They are based on the idea of finding a hyperplane that best separates the features into two classes. <!--SVMs are one of the most robust prediction methods.-->Suppose that we have a set of points in a two-dimensional feature space, with each of the points belonging to one of two classes. An SVM finds what is known as a separating hyperplane: a hyperplane (a line, in the two-dimensional case) which separates the two classes of points from one another.
 
 <p align="center">
-<img src="./img/svmHyperplan.png" height="350" />
+<img src="./img/svmHyperplan.png" height="250" width="250" />
 </p>
 
-Depending on the dataset and how we want to classify it there are three approaches of SVMs: **Hard Margin**, [**Soft Margin**](#soft-margin) and **Kernels**. Since we only have the implementation of Soft Margin approach for the moment, we will expalain only her
+Depending on the dataset and how we want to classify it there are three approaches of SVMs: **Hard Margin**, **Soft Margin** and **Kernels**. We will introduce only the Soft Margin since it is the only implementation we have currently.
+## Soft Margin/Example
 
-## Soft Margin
+Soft Margin consists in classifying the data avoiding overfitting. In other words, we want to allow some misclassifications in the hope of achieving better generality.
+Let's visualize this with an example : 
 
+Given 4 input vectors _x = (x1, x2)_ and 4 output values _y_. 
 
-## Example (maybe remove title)
+```st
+input := #( #( 0 0 ) #( 1 1 ) #( 3 3 ) #( 4 4 ) ).
+output := #( 1 1 -1 -1 ).
+```
 
-For example:
-
-Given 20 input vectors 
+We want to find a hyperplan that separates the two classes each input vector belongs to. SVM were originally designed to only support two-class-problems. So our two classes here are {-1,1}. As you can see, the red points belong to the first class and the blue ones to the second class.
 
 <p align="center">
-<img src="./img/1.png " height="350" /> 
-<img src="./img/2.png " height="350" />
-<img src="./img/3.png " height="350" />
+<img src="./img/1.png " height="250" width="250" />
 </p>
 
+We have an infinit number of possible hyperplans that can separate our data. Since we're looking for the best one we're dealing with an optimazation problem. Therefor, one of the commonly used ways of solving this type of problems is the Stochastic Gradient Descent (SGD) which will be used here. In simple words, SGD is an iterative learning process where an objective function is minimized according to the direction of steepest ascent so that the best coefficients for modeling may be converged upon.
+
+
+<p align="center">
+<img src="./img/2.png " height="250" width="250" /> <img src="./img/White_square_50%_transparency.png" height="250" width="250" />
+<img src="./img/3.png " height="250" width="250" />
+</p>