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
This function was originally proposed as "modified hyperbolic tangent" by Ukrainian scientist Elena V. Soboleva (Russian: Елена В. Соболева) as a utility function for multi-objective optimization and choice modelling in decision-making.
The function has since been introduced into neural network theory and practice.
It was also used in economics for modelling consumption and investment,[1] to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes, to design antenna feeders, and analyze plasma temperatures and densities in the divertor region of fusion reactors.
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. It is worth noting that in this case, the function is not sensitive to flipping the left and right-sides parameters:
The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:
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).