point given by - / > xx - / > where we have (suggestivelydefined :[hx xihxn xi]the important point is that is completely determined by the vector of inner products and the principal eigenvalues and (righteigenvectors of the gram matrix note that each component zm of is of the form zm am, (xi ) ( = the preceding discussion assumed centering of the columns of consider now an then the centered data can be written as en xe where uncentered data matrix en is the matrix of ones consequentlyex en ex ex een en ex een xxx ex ehwhere in > in is the nxn identity matrixormore compactlyxxh and is the vector of ones ex eby [ (xi )ij nto generalize to the kernel settingwe replace and set [ ( ) (xn )]so that lk is the diagonal matrix of the largest eigenvalues of hkh and vk is the corresponding matrix of eigenvectors note that the "usualpca is recovered when we use the linear kernel (xyxy howeverinstead of having only kernels that are explicitly inner products of feature vectorswe are now permitted to implicitly use infinite feature maps (functionsby using kernels example (kernel pcawe simulated pointsx from the uniform distribution on the set ( bc )where br :{(xyr  (diskwith radius rwe apply kernel pca with gaussian kernel (xx exp -kx and compute the functions zm ( ) in ( their density plots are shown in figure the data points are superimposed in each plot from this we see that the principal components identify the radial structure present in the data finallyfigure shows the projections [ (xi ) (xi )] of the original data points onto the first two principal components we see that the projected points can be separated by straight linewhereas this is not possible for the original datasee alsoexample for related problem