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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 functional margin is?

I realize that a similar question has been asked here: How to understand the functional margin in SVM ?

However, the answer given there explains the equation, but not its meaning.

by (33.1k points)

A geometric margin is the Euclidean distance between a certain x (data point) to the hyperplane. 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. so 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.

You should maximize the margin you need more than 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 terms of units of |w|.

Study SVM Algorithms for more details on this. For a training point of view, study Machine Learning Course.