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Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
Neurons also cannot fire faster than a certain rate, motivating sigmoid activation functions whose range is a finite interval. The function looks like () = (+ ′), where is the Heaviside step function.
For example, the Hill coefficient of oxygen binding to haemoglobin (an example of positive cooperativity) falls within the range of 1.7–3.2. [5] <. Negatively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases. =.
A sigmoid curve of an autocatalytic reaction. When t = 0 to 50, the rate of reaction is low. Thereafter, the reaction accelerates, until almost all reactants have been consumed. At that point, the reaction rate tapers off.
If the initial rate of the reaction is measured over a range of substrate concentrations (denoted as [S]), the initial reaction rate increases as [S] increases, as shown on the right. However, as [S] gets higher, the enzyme becomes saturated with substrate and the initial rate reaches V max , the enzyme's maximum rate.
The first examples were the arbitrary width case.George Cybenko in 1989 proved it for sigmoid activation functions. [3] Kurt Hornik [], Maxwell Stinchcombe, and Halbert White showed in 1989 that multilayer feed-forward networks with as few as one hidden layer are universal approximators. [1]
The swish paper was then updated to propose the activation with the learnable parameter β. In 2017, after performing analysis on ImageNet data, researchers from Google indicated that using this function as an activation function in artificial neural networks improves the performance, compared to ReLU and sigmoid functions. [ 1 ]