When using Kernel PCA for dimensionality reduction, is there any simple criterion which can be used to determine the number of components to use?

I am using Kernel PCA with linear kernel, which would be equivalent to normal PCA, but I am using kPCA because my data is extremely sparse but high dimensional, and the number of instances is relatively small compared to the number of dimension. So, centering the data, which PCA requires for computing co-variance, would destroy the sparsity pattern and make computation more difficult.

If I were using PCA, I could plot the eigenvalues of the co-variance matrix in descending order and look for elbow or other methods discussed here. Can I use the same approach with the "centered" kernel matrix?


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The eigenvalues of the centered Gram matrix (centered kernel matrix) are exactly the same as the eigenvalues of the covariance matrix. So whatever method you would want to use with the usual eigenvalues in PCA (e.g. certain proportion of variance explained, or looking for an "elbow" on the "scree plot", etc.), you can use the same method with the kernel eigenvalues.


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