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I've been close reading a paper and was mystified when the author mentioned two kernels: The logistic kernel $\frac{1}{e^x + 2 + e^{-x}}$ and the sigmoid kernel $\frac{2}{\pi}\frac{1}{e^x + e^{-x}}$. I'm mystified because the logistic function, also called the sigmoid function, is $\frac{1}{1+e^{-x}}$, which is a different function that lacks the symmetry of either of those two.

When I asked the author, he pointed to the Wikipedia page called Kernel (statistics). That page does list these two kernels, but I can't find a reference that introduces these two, or why they have these names when they're so different from the logistic/sigmoid function.

Could I have a clearer reference please?

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Kernel is basically a PDF of a respective distribution without parameters. Logistic distribution uses logistic function as CDF, and its PDF would be exactly as you stated in your question (you can check that out by taking a CDF derivative).

I'm pretty curious about the sigmoid example as well, while its CDF is still a sigmoid in a broad sense (DL frameworks mostly stick to a narrow sense where sigmoid=logistic), it looks like a hyperbolic secant distribution, which does not seem that widely used to me to be generalized as a sigmoid kernel.

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