The representer theorem states that any function that takes the form:
f ∗ ∈ arg min f ∈ H k ∑ i = 1 n ℓ ( f ( x i ) , y i ) + Ω ( ‖ f ‖ H k 2 ) {\displaystyle 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 ℓ ( ⋅ ) {\displaystyle \ell (\cdot )} is any loss function and Ω ( ⋅ ) {\displaystyle \Omega (\cdot )} is any regularization term, the optimal solution, f ∗ {\displaystyle {\mathcal {f}}^{*}} , will take the form:
f ∗ = ∑ i = 1 n α i k ( x i , ⋅ ) {\displaystyle {\mathcal {f}}^{*}=\sum _{i=1}^{n}\alpha _{i}k(x_{i},\cdot )}
Where k ( x i , ⋅ ) {\displaystyle k(x_{i},\cdot )} is a kernel.
is any loss function and 
is any regularization term, the optimal solution, 
, will take the form:
is a kernel.
