Why does a separator in a Support Vector Machine always have to be linear?

Question

I just started learning about SVM's today , and found that when the points in the dataset cannot be separated by a boundary that's linear , but rather radial , then we translate that radial boundary to higher dimensions to get a plane.

My question is , why is it not possible to just separate the points in the original dimension using a curved separator ?

Thank you for the help!

Answer 1

1. Isn't the kernel trick doing exactly what you ask? (Linear separator in the higher-dimensional space is representing a curved separator in the original space.)
2. If you are looking for a more "direct" method, you'd need to model your "curved separator," which would require more parameters than a simple linear separator. There's nothing wrong with this (but you'd have to define it from scratch). In fact, this increase in the number of parameters corresponds to why the kernel trick requires working in a higher-dimensional space.