Is it possible to use kernel PCA for feature selection? Is it possible to use kernel principal component analysis (kPCA) for Latent Semantic Indexing (LSI) in the same way as PCA is used?
I perform LSI in R using the prcomp PCA function and extract the features with highest loadings from the first $k$ components. By that I get the features describing the component best.
I tried to use the kpca function (from the kernlib package) but cannot see how to access the weights of the features to a principal component. Is this possible overall when using kernel methods?
 A: I think the answer to your question is negative: it is not possible.
Standard PCA can be used for feature selection, because each principal component is a linear combination of original features, and so one can see which original features contribute most to the most prominent principal components, see e.g. here: Using principal component analysis (PCA) for feature selection.
But in kernel PCA each principal component is a linear combination of features in the target space, and for e.g. Gaussian kernel (which is often used) the target space is infinite-dimensional. So the concept of "loadings" does not really make sense for kPCA, and in fact, kernel principal components are computed directly, bypassing computation of principal axes (which for standard PCA are given in R by prcomp$rotation) altogether, thanks to what is known as kernel trick. See e.g. here: Is Kernel PCA with linear kernel equivalent to standard PCA? for more details. 
So no, it is not possible. At least there is no easy way.
A: The following example (taken from kernlab reference manual) shows you how to access the various components of the kernel PCA:
data(iris)
test <- sample(1:50,20)
kpc <- kpca(~.,data=iris[-test,-5],kernel="rbfdot",kpar=list(sigma=0.2),features=2)

pcv(kpc)        # returns the principal component vectors
eig(kpc)        # returns the eigenvalues
rotated(kpc)    # returns the data projected in the (kernel) pca space
kernelf(kpc)    # returns the kernel used when kpca was performed

Does this answer your question?
