Invert the softmax function

Is it possible to revert the softmax function in order to obtain the original values $x_i$?

$$S_i=\frac{e^{x_i}}{\sum e^{x_i}} $$

In case of 3 input variables this problem boils down to finding $a$, $b$, $c$ given $x$, $y$ and $z$:

\begin{cases} \frac{a}{a+b+c} &= x \\ \frac{b}{a+b+c} &= y \\ \frac{c}{a+b+c} &= z \end{cases}

Is this problem solvable?

Note that in your three equations you must have $x+y+z=1$. The general solution to your three equations are $a=kx$, $b=ky$, and $c=kz$ where $k$ is any scalar.

So if you want to recover $x_i$ from $S_i$, you would note $\sum_i S_i = 1$ which gives the solution $x_i = \log (S_i) + c$ for all $i$, for some constant $c$.

A similar question was asked in a post of reddit. The answer below is adapted from that post:

$S_{i}$ = $\exp(x_{i})/(\sum_{i} \exp(x_{i}))$

Taking ln on both sides:

$\ln(S_{i}) = x_{i} - \ln(\sum_{i} \exp(x_{i}))$

Changing sides:

$x_{i} = \ln(S_{i}) + \ln(\sum_{i} \exp(x_{i}))$

The second term of the right hand side is constant for a particular $i$ and can be written as $C_{i}$. Therefore, we can write:

$x_{i} = \ln(S_{i}) + C$