I'm dealing currently with kernels and kernel PCA. For this purpose I've been reading a few papers on these topics. In this context I've been reading the paper "Kernel Principal Component Analysis" by Schölkopf et. al. While reading the paper certain questions emerged:

  • Performing the kernel PCA, we get a kernel matrix K. From the 'standard' PCA we get Eigenvalues and Eigenvectors. How do we get the Eigenvectors and Eigenvalues from the kernel matrix?

The dual eigenvalue problem is:

$m \cdot \lambda \cdot \alpha = K \cdot \alpha$

where m is the number of data points, alpha are some coefficients and K is our kernel matrix.

  • How are the coefficients alpha determined? What do they stand for?

  • Did I understand correctly that the PCA itself is performed implicitly in the feature space?

  • having the Eigenvalues and Eigenvectors: What is the intuition of them in feature space? What do they mean if we use e.g. a polynomial kernel of degree 2 and obtain the top k-eigenvalues?

  • What is the geometric/visual intuition of the obtained Eigenvalues in feature space?

  • How can we reconstruct from the determined Eigenvalues and Eigenvectors the original dataset?

  • In how far can the Eigenvalues and Eigenvectors be compared to those in the 'standard'/linear PCA?



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