Something that I didn't know, and may help others understand this paper better, is that there's a way to define the sorting a vector through the creative use of the Birkhoff–von Neumann theorem:

https://en.wikipedia.org/wiki/Doubly_stochastic_matrix

which is better explained here:

https://cs.stackexchange.com/questions/4805/sorting-as-a-lin...

where the sorting operation is defined as a linear program. Evidently, this has been known for at least half a century. That said, if a solution to a linear program can be found in a way that's differentiable, this means that the operation of sorting can be found to be differentiable as well. This appears to be trick in the paper and they appear to have a relatively fast way to compute this solution as well, which I think is interesting.