The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF), [nb 1] is a special S-shaped function based on the hyperbolic tangent, given by
| Equation | Left tail control | Right tail control |
|---|---|---|
 |  |
This function was originally proposed as "modified hyperbolic tangent" [nb 1] by Ukrainian scientist Elena V. Soboleva (Елена В. Соболева) as a utility function for multi-objective optimization and choice modelling in decision-making. [1] [2] [3]
The function has since been introduced into neural network theory and practice. [4]
It was also used in economics for modelling consumption and investment, [5] to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes, [6] to design antenna feeders, [7] and analyze plasma temperatures and densities in the divertor region of fusion reactors. [8]
Derivative of the function is defined by the formula:
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. [9] 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 |
|---|---|---|
 |  |
The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:
| Equation | Basic chart | Scaled function |
|---|---|---|
Extremum estimates: |  |  |
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).
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