An ODE characterisation of multi-marginal optimal transport with pairwise cost functions

Luca Nenna; Brendan Pass

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arXiv · 2212.12492

An ODE characterisation of multi-marginal optimal transport with pairwise cost functions

Luca Nenna·Brendan Pass

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Abstract

The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one parameter family of cost functions that interpolates between the original and a special cost function for which the problem's complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $\epsilon$, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge-Kutta schemes to compute solutions to the ODE, and, as a result, the multi-marginal optimal transport problem.

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Luca Nenna, Brendan Pass. 2022-12-23. An ODE characterisation of multi-marginal optimal transport with pairwise cost functions. https://arxiv.org/abs/2212.12492

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