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

TL;DR

This paper demonstrates that the discrete Sinkhorn algorithm converges to a solution of a non-linear parabolic PDE of Monge-Ampere type, enabling efficient approximation of optimal transport maps and distances on manifolds.

Contribution

It establishes a novel connection between the Sinkhorn algorithm and Monge-Ampere equations, providing explicit complexity bounds and practical schemes for optimal transport on manifolds.

Findings

Convergence of Sinkhorn to Monge-Ampere PDE solutions

Explicit complexity bounds for optimal transport algorithms

Efficient schemes for transport on torus and sphere

Abstract

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. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.