The kernel ridge regression was introduced in Section 11.7. Here, it will be restated via its dual representation form. The ridge regression in its primal representation can be cast as

$\begin{array}{ll}\hfill \text{minimize with respect to}\mathit{\theta},\mathit{\xi}& J(\mathit{\theta},\mathit{\xi})=\sum _{n=1}^{N}{\xi}_{n}^{2}+C\parallel \mathit{\theta}{\parallel}^{2},\hfill \\ \hfill \text{subject to}& {y}_{n}-{\mathit{\theta}}^{T}{\mathit{x}}_{n}={\xi}_{n},n=1,2,\dots ,N,\hfill \end{array}$

(11.51)

which leads to the following Lagrangian:

$\begin{array}{l}\hfill L(\mathit{\theta},\mathit{\xi},\mathit{\lambda})=\sum _{n=1}^{N}{\xi}_{n}^{2}+C\parallel \mathit{\theta}{\parallel}^{2}+\sum _{n=1}^{N}{\lambda}_{n}({y}_{n}-{\mathit{\theta}}^{\text{T}}{\mathit{x}}_{n}-{\xi}_{n}),n=1,2,\dots ,N.\end{array}$

(11.52)

Differentiating with respect to θ and ξ_{n}, n = 1,2,…,N, and equating to zero, we obtain

$\begin{array}{l}\hfill \mathit{\theta}=\frac{1}{2C}\sum _{n=1}^{N}{\lambda}_{n}{\mathit{x}}_{n}\end{array}$

(11.53)

and

$\begin{array}{l}\hfill {\xi}_{n}\end{array}$

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