# Understanding the equations for support vectors in SVMs

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.

By definition $$y$$ is labeled as $$\{-1, 1\}$$. A separating hyperplane is one such that
$$\vec{w}^T\vec{x} + b - y > 0$$ when $$y = 1$$
$$\vec{w}^T\vec{x} + b - y < 0$$ when $$y = -1$$
The above inequalities become equalities at the margins of a maximal margin classifier "band", which is what you have shown above. Which class is labeled as $$-1$$ and which as $$1$$ is chosen arbitrary here.