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How can I prove that the exponential $\exp(K)$ of a kernel function $K$ is again a kernel? I think it can be proved using Taylor expansion but I am not sure how.

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Using Taylor expansion around $0$:

$$ \exp(K) = \exp(0) + \exp(0)K + \frac{\exp(0)}{2!}K^2 + \frac{\exp(0)}{3!}K^3 + ...\\ \exp(K) = 1 + K + \frac{1}{2}K^2 + \frac{1}{6}K^3+... $$

we can see that the exponential of a kernel is just an infinite series of multiplications and additions of that kernel.

Using the fact that addition and multiplication of kernels yield valid kernels:

$$ K' = \alpha K_1 + \beta K_2\\ K' = K_1K_2 $$

we can conclude that the exponential of a kernel is a kernel.

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