Radial basis function kernel

In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.^[1]

The RBF kernel on two samples ${\displaystyle \mathbf {x} \in \mathbb {R} ^{k}}$ and ${\displaystyle \mathbf {x'} }$, represented as feature vectors in some input space, is defined as^[2]

${\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {\|\mathbf {x} -\mathbf {x'} \|^{2}}{2\sigma ^{2}}}\right)}$

${\displaystyle \textstyle \|\mathbf {x} -\mathbf {x'} \|^{2}}$ may be recognized as the squared Euclidean distance between the two feature vectors. ${\displaystyle \sigma }$ is a free parameter. An equivalent definition involves a parameter ${\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}}$:

${\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma \|\mathbf {x} -\mathbf {x'} \|^{2})}$

Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure.^[2] The feature space of the kernel has an infinite number of dimensions; for ${\displaystyle \sigma =1}$, its expansion using the multinomial theorem is:^[3]

${\displaystyle {\begin{alignedat}{2}\exp \left(-{\frac {1}{2}}\|\mathbf {x} -\mathbf {x'} \|^{2}\right)&=\exp({\frac {2}{2}}\mathbf {x} ^{\top }\mathbf {x'} -{\frac {1}{2}}\|\mathbf {x} \|^{2}-{\frac {1}{2}}\|\mathbf {x'} \|^{2})\\[5pt]&=\exp(\mathbf {x} ^{\top }\mathbf {x'} )\exp(-{\frac {1}{2}}\|\mathbf {x} \|^{2})\exp(-{\frac {1}{2}}\|\mathbf {x'} \|^{2})\\[5pt]&=\sum _{j=0}^{\infty }{\frac {(\mathbf {x} ^{\top }\mathbf {x'} )^{j}}{j!}}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }\quad \sum _{n_{1}+n_{2}+\dots +n_{k}=j}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right){\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right){\frac {{x'}_{1}^{n_{1}}\cdots {x'}_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\\[5pt]&=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle \end{alignedat}}}$

${\displaystyle \varphi (\mathbf {x} )=\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\left(a_{\ell _{0}}^{(0)},a_{1}^{(1)},\dots ,a_{\ell _{1}}^{(1)},\dots ,a_{1}^{(j)},\dots ,a_{\ell _{j}}^{(j)},\dots \right)}$

where ${\displaystyle \ell _{j}={\tbinom {k+j-1}{j}}}$,

${\displaystyle a_{\ell }^{(j)}={\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\quad |\quad n_{1}+n_{2}+\dots +n_{k}=j\wedge 1\leq \ell \leq \ell _{j}}$