How to smoothly approximate a sign function

Question

I have a function that defined as following

$$f(x) = \begin{cases} 1, & \text{if $x > 0$ } \\ 0, & \text{if $x=0$ } \\ -1, & \text{if $x<0$ } \end{cases}$$ In practice, the $f(x)$ is approximated by a smooth $\tanh(kx)$ or Heaviside function as bellow figure. Could you have other way to represent the $f(x)$ function? What are the benefits of your way?

Update: $\tanh(kx)$ function $k$ controls the smoothness of the sign function. As $k \to \infty$, the function defined in $f(x)=\tanh(kx)$ converges to standard sign function. Similarly, the derivative of $\tanh(x)$ also converges to Dirac delta function as $k \to \infty$. If $k$ is too small, the evolution equation for $x$ acts locally only on a few values around $\{x=0\}$. Hence, the $\tanh(kx)$ function is sensitive with parameter $k$. The parameter $k$ must be chosen carefully.

Hence, My purposed want to reduce/ignore the affect of k, but remains the above smoothly approximation.

Answer 1

You could try $\frac 2 \pi \arctan \lambda x$ for various $\lambda >0$. If you do not tell us what exactly you are looking for, we shall not be able to help.

Answer 2

What you are describing is usually called "sigmoid" functions in the machine learning community, of which some can be found here: http://en.wikipedia.org/wiki/Sigmoid_function

Another way to do it is to perform local integration on the sign function:

$$a_1(x_0) = \frac{1}{2\delta_x}\int_{x_0-\delta_x}^{x_0+\delta_x} s(x)dx$$

And you could do this over and over, recursively:

$$a_n(x_0) = \frac{1}{2\delta_x}\int_{x_0-\delta_x}^{x_0+\delta_x} a_{n-1}(x)dx$$

You would gain one degree of differentiability every time you do so.

Answer 3

If you want something that is perfectly flat but also smooth you should try some smooth transition function like the following example for $n\geq \sqrt{3}$: $$f(x,n) = \begin{cases} -1,\quad x\leq -1;\\ \tanh\left(\frac{nx}{1-x^2} \right),\quad -1<x<1;\\ 1,\quad x\geq 1\end{cases}$$

where you could made it as close as you want to the $\text{sgn}(x)$ function by increasing the value of $n \to \infty$. You could see this function in Desmos.