searching for function composed of s(x) + x with reasonably efficient inverse

Question

To model my data, i need a function that resembles $f(x) = s(x) + x$ with $f : \mathbb{R} \rightarrow \mathbb{R}$, where $s(x)$ is a sigmoid-like function with co-domain $(0,1)$. So if we look at the derivate, a small bump around zero.

The problem now is that in order to efficiently sample the data (which is important for my problem), I need to calculate the inverse of the function.

I would say that the numerical error of the inverse is not as important as efficiency. I am currently approximating the inverse via gauss-newton, but since I need to sample a lot, this turns out to be quite a bottleneck (I am currently doing approx. $53$ iterations). My current $s$ is the standard logistic function.

I have tried a lot of the obvious sigmoid-functions, but I never managed to derive an inverse function.

Edit: $s$ must be twice continuously differentiable.

Answer 1

As you want to compute the inverse, let us start from that end: Consider $$g(x)=\begin{cases}3x^5-10x^3+30x&-1\le x\le 1\\ 15x+8&x\ge1\\ 15x-8&x\le -1\end{cases} $$ Then $g$ is $C^2$ and has no critical points, hence has a $C^2$ inverse $f$ that looks like a line plus a sigmoidal. To match your demands for $x\ll 0$ and $x\gg 0$, we consider a variant $$\tilde g(x):=cg(ax+b)+d$$ for suitable $a,b,c,d$. If I am not mistaken, $ a=\frac{16}{15}$, $b=\frac 8{15}$, $c=\frac1{16}$, $d=0$ should be okay. Also, $\tilde g$ is very efficient to compute (but $s$ perhaps is not).

Answer 2

Not an answer to the question asked

This doesn't provide a function $s$ such that it's sigmoid, but $f(x) = x + s(x)$ is easily invertible. But it still may be of use:

Have you considered an alternative like rejection sampling? That doesn't require the inverse. For something like a sigmoid, this should result in only about a 50% rejection rate, which isn't bad at all. Sure beats $53$ iterations of Gauss-Newton! It also lets you pick a function $s$ that's appropriate for your model, rather than one that's computationally convenient.

In a broader view: are you certain that all the constraints of your problem are necessary? If sampling entails inversion problems, perhaps you can change sampling strategies rather than searching for algebraically convenient functions.