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2.4.2 Centering in feature space
So far, we have assumed that
{()} has zero mean, which is usually not fulfilled. Therefore, the formalism needs to be adjusted (Schölkopf et al., 1998b). The following set of points will be centered:
The above analysis holds if the covariance matrix is computed from
(). Thus, the kernel matrix
K_{ij} = ()^{T}() needs to be replaced by
= ()^{T}(). Using (2.31), can be written as,

= 
()^{T}()  ()^{T}()  ()^{T}() 


+ 
()^{T}() 


= 
K_{ij}  K_{ir}  K_{rj} + K_{rs} . 
(2.32) 
Therefore, we can evaluate the kernel matrix for the centered data using the known matrix . For the remainder of this thesis, I denote with
the eigenvectors of
instead of , and they are normalized according to (2.29) using the eigenvalues of
. The principal components are
= ().
Next: 2.4.3 Common kernel functions
Up: 2.4 Kernel PCA
Previous: 2.4.1 Feature extraction
Heiko Hoffmann
20050322