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Machine Learning by Sergios Theodoridis

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11.9 Kernel Ridge Regression Revisited

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

minimize with respect toθ,ξJ(θ,ξ)=n=1Nξn2+Cθ2,subject toynθTxn=ξn,n=1,2,,N,

si176_e  (11.51)

which leads to the following Lagrangian:

L(θ,ξ,λ)=n=1Nξn2+Cθ2+n=1Nλn(ynθTxnξn),n=1,2,,N.

si177_e  (11.52)

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

θ=12Cn=1Nλnxn

  (11.53)

and

ξn

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