006_Constructing_Kernels

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Section 6.2: Constructing Kernels

One Approach to constructing kernels : choose a feature space mapping latex(${\Phi}(x)$), and then use this to find the kernel (figure 6.1).

$\LARGE $k(x,x^p) = \mathbf{\Phi}(x)^T \mathbf{\Phi}(x^p) =\sum_{i=0}^M {\Phi_{i}}(x) {\Phi_{i}}(x^p) $$

Alternative: construct kernel functions directly. With this we need to validate that the kernel corresponds to a scalar product in the feature space. [http://research.microsoft.com/~cmbishop/PRML/prmlfigs-jpg/Figure6.1a.jpg] [http://research.microsoft.com/~cmbishop/PRML/prmlfigs-jpg/Figure6.1b.jpg] [http://research.microsoft.com/~cmbishop/PRML/prmlfigs-jpg/Figure6.1c.jpg] [http://research.microsoft.com/~cmbishop/PRML/prmlfigs-jpg/Figure6.1d.jpg] [http://research.microsoft.com/~cmbishop/PRML/prmlfigs-jpg/Figure6.1e.jpg] [http://research.microsoft.com/~cmbishop/PRML/prmlfigs-jpg/Figure6.1f.jpg]

$\LARGE $k(\mathbf{x},\mathbf{z}) = ((\mathbf{x}^T \mathbf{z}))^2$$

this leads to:

$\LARGE $k(\mathbf{x},\mathbf{z}) = \mathbf{\Phi(x)^T \Phi(z)} $$

Classes/BMTRY790/KernelMethods/006_Constructing_Kernels (last edited 2008-01-29 16:01:55 by strasbu)