Sigmoid function

Summary

This partial ingest is based on a clipped encyclopedia page. It records sigmoid functions as bounded S-shaped functions, with the logistic function as the canonical example, and notes their roles in neural activation functions, cumulative distribution functions, and growth-curve modeling.

Key Claims

• A sigmoid function has an S-shaped graph, often with output range [0,1] or [-1,1].
• The logistic function is the most common example and is sometimes what “sigmoid” means in neural-network contexts.
• Many common functions are sigmoidal, including logistic, hyperbolic tangent, error-function, arctangent, and some cumulative distribution functions.
• Sigmoid curves are useful in both artificial neurons and saturating growth models because they combine a transition region with limiting asymptotes.

Methods / Formalism

• Logistic sigmoid:

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

• A typical sigmoid is bounded by horizontal asymptotes and has a bell-shaped derivative.
• The logistic sigmoid is invertible on (0,1), with inverse \operatorname{logit}(p)=\log(p/(1-p)).
• The clipped reference emphasizes a broad taxonomy of sigmoid families, but the wiki concept keeps the reusable anchor at the logistic/tanh/CDF level.
• Logistic Sigmoid and Saturating Growth stores the derivative and logistic-growth equations.

Evidence / Experiments

• This is a reference source rather than an experimental paper.
• It is useful as a mathematical and terminology anchor for later neural-network, probability, and growth-model notes.

Connections

• Core reference source for Sigmoid Functions.
• Connects to S-Curves and Saturating Growth because logistic and Gompertz-style curves model bounded growth.
• Logistic Sigmoid and Saturating Growth stores the reusable formal relationship between the logistic sigmoid and logistic growth.
• Connects to Universal Approximation Theorem because classical feed-forward universality results often assume sigmoidal activations.
• Sits in Deep Learning Fundamentals as background on historical activation functions and in Probability and Statistics through CDF-shaped response curves.

Open Questions

• Which sigmoid family is most useful for modeling AI scaling or capability saturation in practice?
• When should a note distinguish carefully between “sigmoid” as a broad function class and “the sigmoid” as the logistic activation?
• How much of the broad taxonomy is worth preserving versus routing to a mathematical annex if sigmoid modeling becomes a recurring topic?

Citation

Wikipedia contributors. (2026). Sigmoid function. Wikipedia.