Kernel Function

Definition

$K:{\mathbb{R}}^p\to \mathbb{R},K({\mathbf{x}}_i,{\mathbf{x}}_j)=⟨\varphi ({\mathbf{x}}_i),\varphi ({\mathbf{x}}_j)⟩$ where $\varphi :{\mathbb{R}}^p\to \mathcal{H}$

The kernel function returns the result of Inner product in $\mathcal{H}$ by using only the original input data.

Examples

Some common kernel functions $K({\mathbf{x}}_i,{\mathbf{x}}_j)$ are as follows:

• Linear: $⟨{\mathbf{x}}_i,{\mathbf{x}}_j⟩$
• Polynomial: $(⟨{\mathbf{x}}_i,{\mathbf{x}}_j⟩+c{)}^{d}c\ge 0,n\ge 1$
• Gaussian radial basis function: $\exp (-\gamma ||{\mathbf{x}}_i-{\mathbf{x}}_j|{|}^2),\gamma >0$
• Sigmoid: $\tanh (\kappa ⟨{\mathbf{x}}_i,{\mathbf{x}}_j⟩+c),\kappa >0,c<0$