# Why is it desirable to have linear separability in SVM?

Ref to above image, clearly a circle can separate the two classes(left image). Why then take so much pain to map it to a function to make it linearly separable (right image) ?

Can anyone please explain ? I really couldn't find anything on the web or youtube lectures on the why

Well, that is the whole idea behind support vector machines! svm are searching for a hyperplane that separates the classes (why the name), and that can of course be done most effectively it the points are linearly separable (that's not a deep point, it is a summary of the full idea). In the example you show, point lie on concentric annular rings, which cannot be separated by any plane, but by introducing a new variable RADIUS---distance from center---you get complete linear separation.

• You mean to say that linear seperability of classes is better/easier than non-linear seperability? Jul 31, 2017 at 15:07
• Classes can be non-linearly separable in an infinite number of ways, so yes, linear separation is clearly easier to handle! And, that is the complete idea behind SVM, so that they work better for data which comply with the asumptions behind the method should hardly be a surprise ... Jul 31, 2017 at 15:09
• @kjetilbhalvorsen I think the key idea behind SVM is kernel trick to save time for computation. But not "using polynomial basis expansion". Jul 31, 2017 at 15:11
• @hdx1101 - The kernel trick makes a lot of stuff feasible, computationally, but that's a big boon to implementation, not the idea behind the method itself. Jul 31, 2017 at 19:45

Why is it desirable to have linear separability in SVM?

SVCs are inherently a linear technique. They find linear boundaries separating (as best possible) different classes. If there is no natural linear boundary for the problem, the choices are either to use a different technique, or to use SVCs with transformed features into a space where there indeed is a linear boundary.

Ref to above image, clearly a circle can separate the two classes(left image). Why then take so much pain to map it to a function to make it linearly separable (right image) ?

This is a classic example. The data classes are separated by a circle, but an SVC cannot find circles directly. However, if the data are transformed using a radial basis function, then in the resulting space, the classes are separated by a linear boundary.