Say I have training data $S_n$ and each point is of the form $x = \langle x_1 , x_2 \rangle$ in the original space (i.e. $x^{(i)} \in \mathbb{R}^2$).
I was considering the following kernel:
$$ K(x,x') = \frac{x^Tx}{|| x ||_{ \mathbb{R}^2 } || x' ||_{ \mathbb{R}^2 } } $$
Which has the following feature vector:
$$ \phi(x) = \frac{x}{|| x ||} $$
I was training to find an intuitive way to draw the decision boundary in the original space compared to the new data space.
This is what it should look like for a simple example:
The trouble that I have is visualizing and understanding rigorously how the decision boundary in the feature space becomes the lines in the original space.
I notice the pattern that is done and where ever the decision boundary intersects the unit circle, is where the line is drawn in the original space.
Somebody have a rigorous justification of this? I can see it in the picture but find it hard to generalize to higher dimensions.