Activation function

Ridge functions are multivariate functions acting on a linear combination of the input variables. Often used examples include:

• Linear activation: ${\displaystyle \phi (\mathbf {v} )=a+\mathbf {v} '\mathbf {b} }$,
• ReLU activation: ${\displaystyle \phi (\mathbf {v} )=\max(0,a+\mathbf {v} '\mathbf {b} )}$,
• Heaviside activation: ${\displaystyle \phi (\mathbf {v} )=1_{a+\mathbf {v} '\mathbf {b} >0}}$,
• Logistic activation: ${\displaystyle \phi (\mathbf {v} )=(1+\exp(-a-\mathbf {v} '\mathbf {b} ))^{-1}}$.

In biologically inspired neural networks, the activation function is usually an abstraction representing the rate of action potential firing in the cell.[7] In its simplest form, this function is binary—that is, either the neuron is firing or not. Neurons also cannot fire faster than a certain rate, motivating sigmoid activation functions whose range is a finite interval.

The function looks like ${\displaystyle \phi (\mathbf {v} )=U(a+\mathbf {v} '\mathbf {b} )}$, where ${\displaystyle U}$ is the Heaviside step function.

If a line has a positive slope, on the other hand, it may reflect the increase in firing rate that occurs as input current increases. Such a function would be of the form ${\displaystyle \phi (\mathbf {v} )=a+\mathbf {v} '\mathbf {b} }$.

Rectified linear unit and Gaussian error linear unit activation functions

A special class of activation functions known as radial basis functions (RBFs) are used in RBF networks, which are extremely efficient as universal function approximators. These activation functions can take many forms, but they are usually found as one of the following functions:

• Gaussian: ${\displaystyle \,\phi (\mathbf {v} )=\exp \left(-{\frac {\|\mathbf {v} -\mathbf {c} \|^{2}}{2\sigma ^{2}}}\right)}$
• Multiquadratics: ${\displaystyle \,\phi (\mathbf {v} )={\sqrt {\|\mathbf {v} -\mathbf {c} \|^{2}+a^{2}}}}$
• Inverse multiquadratics: ${\displaystyle \,\phi (\mathbf {v} )=\left(\|\mathbf {v} -\mathbf {c} \|^{2}+a^{2}\right)^{-{\frac {1}{2}}}}$
• Polyharmonic splines

where ${\displaystyle \mathbf {c} }$ is the vector representing the function center and ${\displaystyle a}$ and ${\displaystyle \sigma }$ are parameters affecting the spread of the radius.

Folding activation functions are extensively used in the pooling layers in convolutional neural networks, and in output layers of multiclass classification networks. These activations perform aggregation over the inputs, such as taking the mean, minimum or maximum. In multiclass classification the softmax activation is often used.

The following table compares the properties of several activation functions that are functions of one fold x from the previous layer or layers:

Name	Plot	Function, ${\displaystyle g(x)}$	Derivative of ${\displaystyle g}$, ${\displaystyle g'(x)}$	Range	Order of continuity
Identity		${\displaystyle x}$	${\displaystyle 1}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{\infty }}$
Binary step		${\displaystyle {\begin{cases}0&{\text{if }}x<0\\1&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle 0}$	${\displaystyle \{0,1\}}$	${\displaystyle C^{-1}}$
Logistic, sigmoid, or soft step		${\displaystyle \sigma (x)\doteq {\frac {1}{1+e^{-x}}}}$	${\displaystyle g(x)(1-g(x))}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$
Hyperbolic tangent (tanh)		${\displaystyle \tanh(x)\doteq {\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}$	${\displaystyle 1-g(x)^{2}}$	${\displaystyle (-1,1)}$	${\displaystyle C^{\infty }}$
Soboleva modified hyperbolic tangent (smht)		${\displaystyle \operatorname {smht} (x)\doteq {\frac {e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}}}$		${\displaystyle (-1,1)}$	${\displaystyle C^{\infty }}$
Rectified linear unit (ReLU)[8]		${\displaystyle {\begin{aligned}(x)^{+}\doteq {}&{\begin{cases}0&{\text{if }}x\leq 0\\x&{\text{if }}x>0\end{cases}}\\={}&\max(0,x)=x{\textbf {1}}_{x>0}\end{aligned}}}$	${\displaystyle {\begin{cases}0&{\text{if }}x<0\\1&{\text{if }}x>0\end{cases}}}$	${\displaystyle [0,\infty )}$	${\displaystyle C^{0}}$
Gaussian Error Linear Unit (GELU)[2]		${\displaystyle {\begin{aligned}&{\frac {1}{2}}x\left(1+{\text{erf}}\left({\frac {x}{\sqrt {2}}}\right)\right)\\{}={}&x\Phi (x)\end{aligned}}}$	${\displaystyle \Phi (x)+x\phi (x)}$	${\displaystyle (-0.17\ldots ,\infty )}$	${\displaystyle C^{\infty }}$
Softplus[9]		${\displaystyle \ln \left(1+e^{x}\right)}$	${\displaystyle {\frac {1}{1+e^{-x}}}}$	${\displaystyle (0,\infty )}$	${\displaystyle C^{\infty }}$
Exponential linear unit (ELU)[10]		${\displaystyle {\begin{cases}\alpha \left(e^{x}-1\right)&{\text{if }}x\leq 0\\x&{\text{if }}x>0\end{cases}}}$ with parameter ${\displaystyle \alpha }$	${\displaystyle {\begin{cases}\alpha e^{x}&{\text{if }}x<0\\1&{\text{if }}x>0\end{cases}}}$	${\displaystyle (-\alpha ,\infty )}$	${\displaystyle {\begin{cases}C^{1}&{\text{if }}\alpha =1\\C^{0}&{\text{otherwise}}\end{cases}}}$
Scaled exponential linear unit (SELU)[11]		${\displaystyle \lambda {\begin{cases}\alpha (e^{x}-1)&{\text{if }}x<0\\x&{\text{if }}x\geq 0\end{cases}}}$ with parameters ${\displaystyle \lambda =1.0507}$ and ${\displaystyle \alpha =1.67326}$	${\displaystyle \lambda {\begin{cases}\alpha e^{x}&{\text{if }}x<0\\1&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle (-\lambda \alpha ,\infty )}$	${\displaystyle C^{0}}$
Leaky rectified linear unit (Leaky ReLU)[12]		${\displaystyle {\begin{cases}0.01x&{\text{if }}x\leq 0\\x&{\text{if }}x>0\end{cases}}}$	${\displaystyle {\begin{cases}0.01&{\text{if }}x<0\\1&{\text{if }}x>0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$
Parametric rectified linear unit (PReLU)[13]		${\displaystyle {\begin{cases}\alpha x&{\text{if }}x<0\\x&{\text{if }}x\geq 0\end{cases}}}$ with parameter ${\displaystyle \alpha }$	${\displaystyle {\begin{cases}\alpha &{\text{if }}x<0\\1&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$
Sigmoid linear unit (SiLU,[2] Sigmoid shrinkage,[14] SiL,[15] or Swish-‍1[16])		${\displaystyle {\frac {x}{1+e^{-x}}}}$	${\displaystyle {\frac {1+e^{-x}+xe^{-x}}{\left(1+e^{-x}\right)^{2}}}}$	${\displaystyle [-0.278\ldots ,\infty )}$	${\displaystyle C^{\infty }}$
Gaussian		${\displaystyle e^{-x^{2}}}$	${\displaystyle -2xe^{-x^{2}}}$	${\displaystyle (0,1]}$	${\displaystyle C^{\infty }}$

The following table lists activation functions that are not functions of a single fold x from the previous layer or layers:

Name	Equation, ${\displaystyle g_{i}\left({\vec {x}}\right)}$	Derivatives, ${\displaystyle {\frac {\partial g_{i}\left({\vec {x}}\right)}{\partial x_{j}}}}$	Range	Order of continuity
Softmax	${\displaystyle {\frac {e^{x_{i}}}{\sum _{j=1}^{J}e^{x_{j}}}}}$    for i = 1, …, J	${\displaystyle g_{i}\left({\vec {x}}\right)\left(\delta _{ij}-g_{j}\left({\vec {x}}\right)\right)}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$
Maxout[17]	${\displaystyle \max _{i}x_{i}}$	${\displaystyle {\begin{cases}1&{\text{if }}j={\underset {i}{\operatorname {argmax} }}\,x_{i}\\0&{\text{if }}j\neq {\underset {i}{\operatorname {argmax} }}\,x_{i}\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$

^ Here, ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

^ For instance, ${\displaystyle j}$ could be iterating through the number of kernels of the previous neural network layer while ${\displaystyle i}$ iterates through the number of kernels of the current layer.

In quantum neural networks programmed on gate-model quantum computers, based on quantum perceptrons instead of variational quantum circuits, the non-linearity of the activation function can be implemented with no need of measuring the output of each perceptron at each layer. The quantum properties loaded within the circuit such as superposition can be preserved by creating the Taylor series of the argument computed by the perceptron itself, with suitable quantum circuits computing the powers up to a wanted approximation degree. Because of the flexibility of such quantum circuits, they can be designed in order to approximate any arbitrary classical activation function.[18]