The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Amp\`ere equations

Understand

We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit.

• The latter evolution equation has previously appeared in different contexts (e.g.
• on the torus it can be be identified with the Ricci flow).
• This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors.