Exercise 12.19: Positive Semidefiniteness of Gaussian Kernel

chapter 12

(a)

Obvious: any finite product and sum of PSD kernels is a PSD kernel.

(b)

Use series expansion of the exponential function and the fact that a pointwise limit of PSD kernels is also a PSD kernel.

(c)

Expand the square and use (b): \begin{equation} k(x, z) = e^{-\tfrac12\norm{x - z}^2} = e^{-\tfrac12\norm{x}^2} e^{-\tfrac12\norm{z}^2} e^{\lra{x, z}^2}. \end{equation}