- - - figure contours of the marginal log-likelihood for the gp regression example the maximum is denoted by cross kernel pca in its basic formkernel pca (principal component analysiscan be thought of as pca in feature space the main motivation for pca introduced in section was as dimensionality reduction technique therethe analysis rested on an svd of the matrix xxp where the data in was first centered via xi, xij where xi = xij what we shall do is to first re-cast the problem in terms of the gram matrix xx[hxi (note the different order of and )and subsequently replace the inner product hxx with (xx for general reproducing kernel to make the linklet us start with an svd of xxudv( the dimensions of xudand are nd dd nand nrespectively then an svd of xx is xx (udv)(udv) (dd)uand an svd of is (udv)(udvv(dd)vlet lr denote the non-zero eigenvalues of xx (orequivalentlyof kand denote the corresponding diagonal matrix by without loss of generality we can assume that the eigenvector of xx corresponding to lk is the -th column of and that the -th column of is an eigenvector of similar to section let uk and vk contain the first columns of and vrespectivelyand let lk be the corresponding kxk submatrix of lk by the svd ( )we have xvk udvvk uk / nextconsider the projection of point onto the -dimensional linear space spanned by the columns of uk -the first principal components we saw in section that this projection simply is the linear mapping > using the fact that uk xvk - / we find that is projected to