denotes dot product, which is simply the elements of the kernel . It seems all that's left is to calculate and normalize the , which can be done by solving the eigenvector equation where is the number of data points in the set, and and are the eigenvalues and eigenvectors of . Then to normalize the eigenvectors , we require that Care must be taken regarding the fact that, whether or not has zero-mean in its original space, it is not guaranteed to be centered in the feature space (which we never compute explicitly). Since centered data is required to perform an effective principal component analysis, we 'centralize' to become where denotes a N-by-N matrix for which each element takes value . We use to perform the kernel PCA algorithm described above. One caveat of kernel PCA should be illustrated here. In linear PCA, we can use the eigenvalues to rank the eigenvectors based on how much of the variation of the data is captured by each principal component. This is useful for data dimensionality reduction and it could also be applied to KPCA. However, in practice there are cases that all variations of the data are same. This is typically caused by a wrong choice of kernel scale. Large datasets In practice, a large data set leads to a large K, and storing K may become a problem. One way to deal with this is to perform clustering on the dataset, and populate the kernel with the means of those clusters. Since even this method may yield a relatively large K, it is common to compute only the top P eigenvalues and eigenvectors of the eigenvalues are calculated in this way. Example Consider three concentric clouds of points (shown); we wish to use kernel PCA to identify these groups. The color of the points does not represent information involved in the algorithm, but only shows how the transformation relocates the data points. First, consider the kernel Applying this to kernel PCA yields the next image. Now consider a Gaussian kernel: That is, this kernel is a measure of closeness, equal to 1 when the points coincide and equal to 0 at infinity. Note in particular that the first principal component is enough to distinguish the three different groups, which is impossible using only linear PCA, because linear PCA operates only in the given (in this case two-dimensional) space, in