Suppose I'm given a kernel,

$$ K(x,y) : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $$

In order to describe/understand the (unique) associated RKHS, I seek its eigenfunctions, as per Mercer's theorem, which should give a "feel" for the space. How do I do this?

  • For generality, I want a numerical procedure.
  • Presumably, this involves discretising a part of $\mathbb{R}$ and using eig or eigh from Numpy or Matlab.
  • But I guess I need to know the inner product first. How? (Numerically, again, is ok).
  • What if, as for the Gaussian (RBF) kernel, $K(x,y) = e^{-\frac{\|x - y\|^2}{2\sigma^2}}$, the inner product is defined in Fourier space (see p28)

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