Soboleva modified hyperbolic tangent

Template:Short description Template:Use dmy dates Template:Use list-defined references The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF), is a special S-shaped function based on the hyperbolic tangent, given by

Equation	Left tail control	Right tail control
$\mathrm{smht}x=\frac{e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}.$

History

This function was originally proposed as "modified hyperbolic tangent" by Ukrainian scientist Elena V. Soboleva (Template:Lang) as a utility function for multi-objective optimization and choice modelling in decision-making.

Practical usage

The function has since been introduced into neural network theory and practice.

It was also used in economics for modelling consumption and investment, to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes, to design antenna feeders,Template:Pred and analyze plasma temperatures and densities in the divertor region of fusion reactors.

Sensitivity to parameters

Derivative of the function is defined by the formula:

$${\mathrm{smht}}^{\prime }(x)\doteq \frac{a{e}^{ax}+b{e}^{-bx}}{e^{cx}+e^{-dx}}-\mathrm{smht}(x)\frac{c{e}^{cx}-d{e}^{-dx}}{e^{cx}+e^{-dx}}$$

The following conditions are keeping the function limited on y-axes: a ≤ c, b ≤ d.

A family of recurrence-generated parametric Soboleva modified hyperbolic tangent activation functions (NPSMHTAF, FPSMHTAF) was studied with parameters a = c and b = d. It is worth noting that in this case, the function is not sensitive to flipping the left and right-sides parameters:

Equation	Left prevalence	Right prevalence
$\frac{e^{ax}-e^{-bx}}{e^{ax}+e^{-bx}}=\frac{e^{bx}-e^{-ax}}{e^{bx}+e^{-ax}}$

The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:

Equation	Basic chart	Scaled function
$\mathrm{ssht}x=\frac{e^x-e^{-x}}{e^{\alpha x}+e^{-\beta x}}.$ Extremum estimates: $x_{\min }=\frac{1}{2}\ln \frac{\beta -1}{\beta +1};$ $x_{\max }=\frac{1}{2}\ln \frac{\alpha +1}{\alpha -1}.$

With parameters a = b = c = d = 1 the modified hyperbolic tangent function reduces to the conventional tanh(x) function, whereas for a = b = 1 and c = d = 0, the term becomes equal to sinh(x).

See also

• Activation function
• e (mathematical constant)
• Equal incircles theorem, based on sinh
• Hausdorff distance
• Inverse hyperbolic functions
• List of integrals of hyperbolic functions
• Poinsot's spirals
• Sigmoid function

Further reading

• Template:Cite journal (20 pages) [1]