From the perspective of fundamental science sigmoid functions are of special interest in abstract areas such as approximation theory, functional analysis and probability theory. More speci cally, sigmoid function are an object of interest in Hausdor approximations, fuzzy set theory, cumulative distribution functions, impulsive functions, etc.

Apr 18, 2021 · The idea/intuition behind this crucial construction is that $\sigma(x)$ can be seen as a smoothed version of a step function (by the way, $\sigma$ or the true step function mimics the behavior of a neuron in the brain).

In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) hp-mesh refinement for solution of PDEs, and (4) double exponential (DE) and generalized...

In this paper, we compared seven ways in total for several degrees of polynomials to approximate a commonly used sigmoid function in neural networks. We focused on the approximation errors beyond the domain used to approximate, which have …

positions of a fixed, univariate function and a set ofaffine functionals can uniformly approximate any continuous function of n real variables with support in the unit hypercube; only mild conditions are imposed on the univariate function.

Aug 1, 2009 · Our aim, in this paper, is to introduce and study approximation operators with the function ϕ (x), i.e., the neural network operators and quasi-interpolation operators. Using these operators as approximation tools, we will consider the estimates of the rate of approximating continuous functions.

Mar 1, 2017 · In this work we study the behaviour of three families of sigmoid functions whenever their slope increases. From practical point of view we may need to know when the steep part of a sigmoid function can be assumed as instantaneous.

Aug 1, 2015 · We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the Hyper-log–logistic function.

In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) h p -mesh refinement for solution of \pdes, and (4) double exponential (DE) and generali...

To substitute a sigmoid function by a step function (or conversely) we need to know the approximation error between the two functions. A natural metric used in such a situation is the Hausdorff metric between the graphs of the functions.