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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.

Andreas G.
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    You almost have it! You surely need the Taylor expansion (https://en.wikipedia.org/wiki/Matrix_exponential). You need to prove that is positive definite. If A > 0, are its powers positive? – jpmuc Dec 29 '17 at 12:04
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    everything you need is in Dougal's answer here https://stats.stackexchange.com/questions/35634/how-to-prove-that-the-radial-basis-function-is-a-kernel/150964#150964 – jld Dec 29 '17 at 14:33

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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.

Andreas G.
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