I am struggling with this point (since my Math is abysmal). I have looked at various YouTube videos (most of them just gloss over this or the Math is beyond me) and read various blog posts (same problem).
Basically, all I want to understand is: Why do the equations for the hyperplanes that the support vectors are on look like this?
$\vec{w}^T\vec{x}+b=1$
$\vec{w}^T\vec{x}+b=-1$
Why 1/-1?
I understand that for points on the hyperplane, this is true:
$\vec{w}^T\vec{x}+b=0$
For points "above" or in the direction of $\vec{w}$ it's $ > 0 $, for the other case it's $ < 0 $
I just don't really understand where the 1/-1 come from. Form what I have read, it has to do with scaling and ultimately choosing 1 is arbitrary, but I don't get it.
What is the problem with the scaling and I kind of makes the optimization problem easier, but the penny just won't drop.
I understand that by doing $c\vec{w}^T\vec{x}+cb=0$, I get the same hyperplane, but I don't understand why because of that I can just go: $\vec{w}^T\vec{x}+b=1$.
So in my head I am picturing this:
The equation for the support vectors are $\vec{w}^T\vec{x}+b= +d / -d $. I basically make two copies of my hyperplane and move them upwards/downwards until each copy hits the first data point and then I place my hyperplane in the middle, so it is equidistant from the other two hyperplanes. Then my hyperplane is the one with the maximum margin.
But I still don't understand why it is okay to just say that d = 1
.