Soboleva modified hyperbolic tangent

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
${\displaystyle \operatorname {smht} x={\frac {e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}}.}$

History

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]

Practical usage

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][predatory publisher]} and analyze plasma temperatures and densities in the divertor region of fusion reactors.^[8]

Sensitivity to parameters

Derivative of the function is defined by the formula:

$${\displaystyle \operatorname {smht} '(x)\doteq {\frac {ae^{ax}+be^{-bx}}{e^{cx}+e^{-dx}}}-\operatorname {smht} (x){\frac {ce^{cx}-de^{-dx}}{e^{cx}+e^{-dx}}}}$$

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.^[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
${\displaystyle {\frac {e^{ax}-e^{-bx}}{e^{ax}+e^{-bx}}}={\frac {e^{bx}-e^{-ax}}{e^{bx}+e^{-ax}}}}$

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

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

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

Notes

1. Soboleva proposed the name "modified hyperbolic tangent" (mtanh, mth), but since other authors used this name also for other functions, some authors have started to refer to this function as "Soboleva modified hyperbolic tangent".

References

1. Soboleva, Elena Vladimirovna; Beskorovainyi, Vladimir Valentinovich (2008). The utility function in problems of structural optimization of distributed objects Функция для оценки полезности альтернатив в задачах структурной оптимизации территориально распределенных объектов. Четверта наукова конференція Харківського університету Повітряних Сил імені Івана Кожедуба, 16–17 квітня 2008 (The fourth scientific conference of the Ivan Kozhedub Kharkiv University of Air Forces, 16–17 April 2008) (in Russian). Kharkiv, Ukraine: Kharkiv University of Air Force (HUPS/ХУПС). p. 121.
2. Soboleva, Elena Vladimirovna (2009). S-образная функция полезности част-ных критериев для многофакторной оценки проектных решений [The S-shaped utility function of individual criteria for multi-objective decision-making in design]. Материалы XIII Международного молодежного форума «Радиоэлектро-ника и молодежь в XXI веке» (Materials of the 13th international youth forum "Radioelectronics and youth in the 21st century") (in Russian). Kharkiv, Ukraine: Kharkiv National University of Radioelectronics (KNURE/ХНУРЕ). p. 247.
3. Beskorovainyi, Vladimir Valentinovich; Soboleva, Elena Vladimirovna (2010). ИДЕНТИФИКАЦИЯ ЧАСТНОй ПОлЕЗНОСТИ МНОГОФАКТОРНЫХ АлЬТЕРНАТИВ С ПОМОЩЬЮ S-ОБРАЗНЫХ ФУНКЦИй [Identification of utility functions in multi-objective choice modelling by using S-shaped functions] (PDF). Problemy Bioniki: Respublikanskij Mežvedomstvennyj Naučno-Techničeskij Sbornik БИОНИКА ИНТЕЛЛЕКТА [Bionics of Intelligence] (in Russian). Vol. 72, no. 1. Kharkiv National University of Radioelectronics (KNURE/ХНУРЕ). pp. 50–54. ISSN 0555-2656. UDK 519.688: 004.896. Archived (PDF) from the original on 2022-06-21. Retrieved 2020-06-19. (5 pages)
4. Malinova, Anna; Golev, Angel; Iliev, Anton; Kyurkchiev, Nikolay (August 2017). "A Family Of Recurrence Generating Activation Functions Based On Gudermann Function" (PDF). International Journal of Engineering Researches and Management Studies. 4 (8). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria: 38–48. ISSN 2394-7659. Archived (PDF) from the original on 2022-07-14. Retrieved 2020-06-19. (11 pages)
5. Orlando, Giuseppe (2016-07-01). "A discrete mathematical model for chaotic dynamics in economics: Kaldor's model on business cycle". Mathematics and Computers in Simulation. 8th Workshop STRUCTURAL DYNAMICAL SYSTEMS: Computational Aspects; Edited by Nicoletta Del Buono, Roberto Garrappa and Giulia Spaletta and Nonstandard Applications of Computer Algebra (ACA’2013); Edited by Francisco Botana, Antonio Hernando, Eugenio Roanes-Lozano and Michael J. Wester. 125: 83–98. doi:10.1016/j.matcom.2016.01.001. ISSN 0378-4754.
6. Tuev, Vasily I.; Uzhanin, Maxim V. (2009). ПРИМЕНЕНИЕ МОДИФИЦИРОВАННОЙ ФУНКЦИИ ГИПЕРБОЛИЧЕСКОГО ТАНГЕНСА ДЛЯ АППРОКСИМАЦИИ ВОЛЬТАМПЕРНЫХ ХАРАКТЕРИСТИК ПОЛЕВЫХ ТРАНЗИСТОРОВ [Using modified hyperbolic tangent function to approximate the current-voltage characteristics of field-effect transistors] (in Russian). Tomsk, Russia: Tomsk Politehnic University (TPU/ТПУ). pp. 135–138. No. 4/314. Archived from the original on 2017-08-15. Retrieved 2015-11-05. (4 pages)
7. Golev, Angel; Djamiykov, Todor; Kyurkchiev, Nikolay (2017-11-23) [2017-10-09, 2017-08-19]. "Sigmoidal Functions In Antenna-Feeder Technique" (PDF). International Journal of Pure and Applied Mathematics. 116 (4). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria / Technical University of Sofia, Sofia, Bulgaria: Academic Publications, Ltd.: 1081–1092. doi:10.12732/ijpam.v116i4.23 (inactive 2024-11-01). ISSN 1311-8080. Archived (PDF) from the original on 2020-06-19. Retrieved 2020-06-19.{{cite journal}}: CS1 maint: DOI inactive as of November 2024 (link) (12 pages)
8. Rubino, Giulio (2018-01-15) [2018-01-14]. Power Exhaust Data Analysis and Modeling Of Advanced Divertor Configuration (Thesis). Joint Research Doctorate In Fusion Science And Engineering Cycle XXX (in English, Italian, and Portuguese). Padova, Italy: Centro Ricerche Fusione (CRF), Università degli Studi di Padova / Università degli Studi di Napoli Federico II / Instituto Superior Técnico (IST), Universidade de Lisboa. p. 84. ID 10811. Archived from the original on 2020-06-19. Retrieved 2020-06-19.{{cite book}}: CS1 maint: bot: original URL status unknown (link) (2+viii+3*iii+102 pages)
9. Golev, Angel; Iliev, Anton; Kyurkchiev, Nikolay (June 2017). "A Note on the Soboleva' Modified Hyperbolic Tangent Activation Function" (PDF). International Journal of Innovative Science, Engineering & Technology (JISET). 4 (6). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria: 177–182. ISSN 2348-7968. Archived (PDF) from the original on 2020-06-19. Retrieved 2020-06-19. (6 pages)

Further reading

• Iliev, Anton; Kyurkchiev, Nikolay; Markov, Svetoslav (2017). "A Note on the New Activation Function of Gompertz Type". Biomath Communications. 4 (2). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria / Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria: Biomath Forum (BF). doi:10.11145/10.11145/bmc.2017.10.201. ISSN 2367-5233. Archived from the original on 2020-06-20. Retrieved 2020-06-19. (20 pages)