# Soft Exponential Activation Function

A Soft Exponential Activation Function is a parametric neuron activation function that is based on the inperpolation of the identity, logarithm and exponential functions .

## References

### 2018

Name Plot Equation Derivative (with respect to x) Range Order of continuity Monotonic Derivative Monotonic Approximates identity near the origin
(...) (...) (...) (...) (...) (...) (...) (...) (...)
SoftExponential [1] $f(\alpha,x) = \begin{cases} -\frac{\ln(1-\alpha (x + \alpha))}{\alpha} & \text{for } \alpha \lt 0\\ x & \text{for } \alpha = 0\\ \frac{e^{\alpha x} - 1}{\alpha} + \alpha & \text{for } \alpha \gt 0\end{cases}$ $f'(\alpha,x) = \begin{cases} \frac{1}{1-\alpha (\alpha + x)} & \text{for } \alpha \lt 0\\ e^{\alpha x} & \text{for } \alpha \ge 0\end{cases}$ $(-\infty,\infty)$ $C^\infty$ Yes Yes Template:Depends
Sinusoid[2] $f(x)=\sin(x)$ $f'(x)=\cos(x)$ $[-1,1]$ $C^\infty$ No No Yes
Sinc $f(x)=\begin{cases} 1 & \text{for } x = 0\\ \frac{\sin(x)}{x} & \text{for } x \ne 0\end{cases}$ $f'(x)=\begin{cases} 0 & \text{for } x = 0\\ \frac{\cos(x)}{x} - \frac{\sin(x)}{x^2} & \text{for } x \ne 0\end{cases}$ $[\approx-.217234,1]$ $C^\infty$ No No No
Gaussian $f(x)=e^{-x^2}$ $f'(x)=-2xe^{-x^2}$ $(0,1]$ $C^\infty$ No No No

Here, H is the Heaviside step function.

α is a stochastic variable sampled from a uniform distribution at training time and fixed to the expectation value of the distribution at test time.

### 2015

1. Godfrey, Luke B.; Gashler, Michael S. (2016-02-03). "A continuum among logarithmic, linear, and exponential functions, and its potential to improve generalization in neural networks". 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management: KDIR 1602: 481–486. arXiv:1602.01321. Bibcode 2016arXiv160201321G.
2. Gashler, Michael S.; Ashmore, Stephen C. (2014-05-09). “Training Deep Fourier Neural Networks To Fit Time-Series Data". arXiv:1405.2262 Freely accessible cs.NE.