Radial Basis Function Kernel

Analyses the similarity of data points (more or less Clustering).

Pros

• most popular kernel
• useful when data is arranged in clusters

Cons

• only good in low-dimensional spaces $\frac{v_r^n}{v_1^n}=\frac{c\cdot r^n}{c\cdot 1^n}=r^n\underset{r<1}{\to }0$

Roughly, a Similarity function between a pair of examples, via the minus sign. Uses exponential function to map values between 1 (similar) and 0 (dissimilar).

$$\kappa \left(x^{(i)},x^{(j)}\right)=\exp \left(-\frac{{\Vert x^{(i)}-x^{(j)}\Vert }^2}{2{\sigma }^2}\right)$$

or simplified to

$$\kappa \left(x^{(i)},x^{(j)}\right)=\exp \left(-\gamma {\Vert x^{(i)}-x^{(j)}\Vert }^2\right).$$

Here $\gamma =\frac{1}{2{\sigma }^2}$ is a free parameter that can be optimized and set in the scikit learn model. You can control the tightness of the fit with this parameter which of course also affects Overfitting and Underfitting of the model.

Low Gamma Value

High Gamma Value