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Solutions for w and a

If we take the gradient of J in terms of w, we can solve for w to get

\LARGE $$\mathbf{w} = -\frac{1}{\lambda} \sum_{n=1}^{N}\{\mathbf{w}^{T} \phi \left( \mathbf{x}_{n}\right) -t_{n}\} \phi(\mathbf{x}_{n})= \sum_{n=1}^{N}\{a_{n}\phi(\mathbf{x}_{n})\}= \mathbf{\Phi}^T\mathbf{a}$$

Now consider the expression for J in terms of a and take the gradient

\LARGE $\mathbf{a} = (\mathbf{K} + \lambda\mathbf{I}_{N})^{-1}\mathbf{t}$

Linear regression in terms of the kernel

Substitute the solution for a back into the linear regression model

\LARGE $$y(\mathbf{x}) = \mathbf{w}^T\phi(\mathbf{x}) = \mathbf{a}^{T}\mathbf{\Phi}\phi(\mathbf{x}) = \mathbf{k}(\mathbf{x})^{T}(\mathbf{K} + \lambda\mathbf{I}_{N})^{-1}\mathbf{t}$$

Classes/BMTRY790/KernelMethods/002_Linear_Regression_Dual_Formulation (last edited 2008-01-29 15:17:38 by mullerb)