SVM - what is a functional margin?

Question

A geometric margin is simply the euclidean distance between a certain x (data point) to the hyperlane.

What is the intuitive explanation to what a function

Answer 1

"A geometric margin is simply the euclidean distance between a certain x (data point) to the hyperlane. "

I don't think that is a proper definition for the geometric margin, and I believe that is what is confusing you. The geometric margin is just a scaled version of the functional margin.

You can think the functional margin, just as a testing function that will tell you whether a particular point is properly classified or not. And the geometric margin is functional margin scaled by ||w||

If you check the formula:

You can notice that independently of the label, the result would be positive for properly classified points (e.g sig(1*5)=1 and sig(-1*-5)=1) and negative otherwise. If you scale that by ||w|| then you will have the geometric margin.

Why does the geometric margin exists?

Well to maximize the margin you need more that just the sign, you need to have a notion of magnitude, the functional margin would give you a number but without a reference you can't tell if the point is actually far away or close to the decision plane. The geometric margin is telling you not only if the point is properly classified or not, but the magnitude of that distance in term of units of |w|