Invert the softmax function

Question

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?

Answer 1

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 $c$.

Answer 2

The softmax function is defined as:

$$S_i = \frac{\exp(x_i)}{\sum_{j} \exp(x_j)}$$

Taking the natural logarithm of both sides:

$$\ln(S_i) = x_i - \ln(\sum_{j} \exp(x_j))$$

Rearranging the equation:

$$x_i = \ln(S_i) + \ln(\sum_{j} \exp(x_j))$$

The second term on the right-hand side is a constant over all $i$ and can be written as $C$. Therefore, we can write:

$$x_i = \ln(S_i) + C$$

This answer is adapted from this post on Reddit.