Regarding the Differentiability Issue of the SoftSort Operator

[MarsMaxwell]

The calculation formula of the softsort operator is as follows:

$$Soft Sort _{\tau}^{d}(s)=softmax\left(\frac{-d\left(sort(s) \mathbb{1}^{T}, \mathbb{1} s^{T}\right)}{\tau}\right)$$

However, the $sort(s)$ operation is used in the formula. According to the description in the paper, isn’t $sort(s)$ non-differentiable? From the perspectives of mathematical principles and machine learning engineering implementation, how can this be explained?

[sprillo]

Hi @MarsMaxwell ! sort(s) is continuous, differentiable almost everywhere, and provides meaningful gradients, much like the ReLU function. The problem is argsort(s), not sort(s). As per the abstract: argsort(s) "has a discrete image and thus zero gradients almost everywhere". Plot these two functions when $s$ is two-dimensional and you will see what's going on.

Best,
Sebastian