Talk:Activation function

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What does "${\displaystyle f(x)\approx x}$ when ${\displaystyle x\approx 0}$" mean?

What is the purpose of the column "${\displaystyle f(x)\approx x}$ when ${\displaystyle x\approx 0}$"? It's unclear what the expression means, as I wrote in the {{clarify}} template. What I don't understand are:

• By substitution, is this equivalent to the expression ${\displaystyle f(x)\approx 0}$ when ${\displaystyle x\approx 0}$?
• If not, does ${\displaystyle f'(x)\approx 1}$ when ${\displaystyle x\approx 0}$?
• Does ${\displaystyle f(x)}$ have to pass through the origin?

These ambiguities apparently lead to some disagreement to what should actually be the truth value of that expression (yes or no?). I think we need to answer these questions, otherwise I don't know if it makes any sense to keep the column. —Kri (talk) 11:56, 5 May 2016 (UTC)

• It's a vague approximation to ${\displaystyle f(x)\approx 0}$, but ${\displaystyle f(x)\approx x}$ is closer. For one thing, ${\displaystyle f(x)\approx 0}$ would imply ${\displaystyle f'(x)\approx 0}$ rather than ${\displaystyle f'(x)\approx 1}$
• We know little of the derivative because we're now stacking approximations on top of each other. In particular it can become risky to assume much about the range of x for where this assumption holds usefully true. Although, yes, ${\displaystyle f'(x)\approx 1}$
• Yes
Andy Dingley (talk) 12:27, 5 May 2016 (UTC)
Okay, so if I understand you corectly, saying that ${\displaystyle f(x)\approx 0}$ when ${\displaystyle x\approx 0}$ for an activation function ${\displaystyle f}$ is equivalent to saying that ${\displaystyle f'(x)\approx 1}$ when ${\displaystyle x\approx 0}$ and that ${\displaystyle f(x)}$ passes through the origin, or more formally that ${\displaystyle f(0)=0}$ and ${\displaystyle f'(0)=1}$; is that correct? —Kri (talk) 21:58, 7 May 2016 (UTC)