Representer theorem

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

$f^{*}\in \arg \underset{f\in {\mathscr{H}}_{𝓀}}{\min }\sum _{i=1}^n\ell (f(x_i),y_i)+\mathrm{\Omega }({\Vert f\Vert }_{{\mathscr{H}}_{𝓀}}^2)$

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

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

Where $k(x_i,\cdot )$ is a kernel.