12.1Centering Data in Feature Space

It is in fact very difficult to explicitly center the data in feature space. But, we know that the final algorithm only depends on the kernel matrix, so if we can center the kernel matrix we are done as well. A kernel matrix is given byK__ij= ΦiΦT__j. We now center the features using,

(12.11)

CHAPTER12. KERNELPRINCIPALCOMPONENTSANALYSIS

Hence the kernel in terms of the new features is given by,

Hence, we can compute the centered kernel in terms of the non-centered kernel alone and no features need to be accessed.

At test-time we need to compute,

(12.17)

Using a similar calculation (left for the reader) you can find that this can be expressed easily in terms ofK(ti__,xj)andK(xi__,xj)as follows,

(12.18)

Chapter 13