zmVault/sigmoid-functions.md

id, aliases, tags, title

id	aliases	tags	title
		authorship/original destiny/permanent status/incomplete topic/math type/encyclopedia	Sigmoid Functions

Sigmoid Functions

!sigmoid-function-example.excalidraw

f'(x) is a sigmoid function.

A sigmoid function compresses a function with infinite range to (0,1)

This is preferable to a floor-ceiling piece-wise function since it remains differentiable.

Formula

\sigma(x) = \frac{1}{1 + e^{-x}}

\begin{document}
  \begin{tikzpicture}[domain=-5:5]
    \draw[->] (-5.2,0) -- (5.2,0) node[right] {$x$}; % x axis
    \draw[->] (0,-5.2) -- (0,5.2) node[above] {$y$}; % y axis
    \draw[color=red]    plot (\x,\x)               node[right] {$f(x) = x$};
    \draw[color=orange] plot (\x,{1/(1+exp(-\x))}) node[right] {$\sigma(x) = \frac{1}{1 + e^{-x}}$};
  \end{tikzpicture}
\end{document}