Representer theorem

The representer theorem states that any function that takes the form:

$f^{*}\in \arg \min _{f\in {\mathcal {H_{k}}}}\sum _{i=1}^{n}\ell (f(x_{i}),y_{i})+\Omega ({\Vert f\Vert }_{\mathcal {H_{k}}}^{2})$

where $\ell (\cdot )$ is any loss function and $\Omega (\cdot )$ is any regularization term, the optimal solution, ${\mathcal {f}}^{*}$, will take the form:

${\mathcal {f}}^{*}=\sum _{i=1}^{n}\alpha _{i}k(x_{i},\cdot )$

Where $k(x_{i},\cdot )$ is a kernel.