Hyperplanes and Support Vector Machines I have the following question regarding support vector machines: So we are given a set of
training points $\{x_i\}$ and a set of binary labels $\{y_i\}$.
Now usually the hyperplane classifying the points is defined as:
$ w \cdot x + b = 0 $
First question: Here $x$ does not denote the points of the training set, but the points on the separating (hyper)plane, right?
In a next step, the lecture notes state that the function:
$f(x) = \mathrm{sign}\, (w \cdot x + b)$
correctly classify the training data.
Second question: Now I don't understand that, since it was stated earlier that $ w \cdot x + b = 0 $, so how can something which is defined to be zero have a sign?
Two additonal questions:
(1) You have added that a slack variable might be introduced for non-linearly separable data - how do we know that the data is not linearly separable, as far as I understand the mapping via kernel has as a purpose to map the data into a vector space where in the optimal case it would be linearly separable (and why not using a non-linear discriminant function altogether instead of introducing a slack variable?)
(2) I've seen that for the optimization only one of the inequalities $w \cdot x = b \geqslant 1$ is being used as a linear constrained - why?

For the first equation, $w\cdot x+b=0$, $w$ is the direction normal (orthogonal/perpendicular) to the hyperplane. You are correct that the $x$ (satisfying this equation) are the points on the hyperplane.
In the second equation, the $x$ are the training (or test) data. These should not lie on the hyperplane, where $f(x)=0$, but should lie on either side of the hyperplane - since, by construction, the hyperplane separates the two classes.
Points that lie on the hyperplane, where $f(x)=0$, will not be able to be classified.
