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I'm trying to understand a paper about the mathematics behind the Word2Vec model. I have come across an expression that I'm sure it's obvious but I have not been able to quite understand on my own. These are the facts we have:

$$ y = f(u)=\sigma(u)= \frac{1}{1+e^{-u}} $$

where $\sigma$ it's the sigmoid function with the following properties:

$$ \sigma(-x)=1-\sigma(x), \,\,\,\,\,\,\, \frac{d\sigma(x)}{dx}=\sigma(x)\sigma(-x) $$

Taking that into consideration, I need to compute $\frac{\partial y}{\partial u}$, and the paper says the following:

$$ \frac{\partial y}{\partial u} = y(1-y) $$

Can anyone explain the steps there, please? This is the paper I'm referring to, in case I have missed something else needed. Thanks!

J.G.
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Antonio Gamiz Delgado
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  • Are you allowed to use the properties you stated without proving them? (Or for that matter, have you proven them already?) Because they trivially imply $\sigma^\prime=\sigma(1-\sigma)$. – J.G. Mar 02 '22 at 19:59
  • Yeah, I can use them without proving them. Using the $\sigma'$ notation I have already seen it, the $d$ notation got me a little bit confused. Thanks! – Antonio Gamiz Delgado Mar 02 '22 at 20:04

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