  The sorting operation is one of the most commonly used building blocks in
computer programming. In machine learning, it is often used for robust
statistics. However, seen as a function, it is piecewise linear and as a result
includes many kinks where it is non-differentiable. More problematic is the
related ranking operator, often used for order statistics and ranking metrics.
It is a piecewise constant function, meaning that its derivatives are null or
undefined. While numerous works have proposed differentiable proxies to sorting
and ranking, they do not achieve the $O(n \log n)$ time complexity one would
expect from sorting and ranking operations. In this paper, we propose the first
differentiable sorting and ranking operators with $O(n \log n)$ time and $O(n)$
space complexity. Our proposal in addition enjoys exact computation and
differentiation. We achieve this feat by constructing differentiable operators
as projections onto the permutahedron, the convex hull of permutations, and
using a reduction to isotonic optimization. Empirically, we confirm that our
approach is an order of magnitude faster than existing approaches and showcase
two novel applications: differentiable Spearman's rank correlation coefficient
and least trimmed squares.
