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] [ predatory publisher ] 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).
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
A logistic function or logistic curve is a common S-shaped curve with the equation
A sigmoid function refers specifically to a function whose graph follows the logistic function. It is defined by the formula:
In mathematics, a cubic function is a function of the form that is, a polynomial function of degree three. In many texts, the coefficientsa, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division. This is in contrast to an algebraic function.
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and motion.
In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.
The International Mathematics Competition (IMC) for University Students is an annual mathematics competition open to all undergraduate students of mathematics. Participating students are expected to be at most twenty three years of age at the time of the IMC. The IMC is primarily a competition for individuals, although most participating universities select and send one or more teams of students. The working language is English.
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.
The University of Plovdiv "Paisii Hilendarski", also known as The Paisii Hilendarski University of Plovdiv, is a university located in Plovdiv, Bulgaria. It was founded in 1962 and has nine faculties.
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.
Hilmi İbar is professor of chemistry, former dean of the pedagogic faculty and former head of the International Office at the Trakya University in Edirne. He is also vice president of the Balkan Universities Network.
In the context of artificial neural networks, the rectifier or ReLU activation function is an activation function defined as the non-negative part of its argument:
Dink is a village in Southern Bulgaria, Maritsa Municipality, Plovdiv Province. As of 15 June 2020, the village has a population of 909.
{{cite journal}}: CS1 maint: DOI inactive as of November 2024 (link) (12 pages){{cite book}}: CS1 maint: bot: original URL status unknown (link) (2+viii+3*iii+102 pages)