An ordinary differential equation for entropic optimal transport and its linearly constrained variants

TL;DR

This work develops an ODE-based characterization of entropically regularized optimal transport with discrete marginals and linear constraints, extending to multi-marginal and constrained variants. By introducing a dual potential $\varphi$ and deriving the ODE $\frac{d\varphi}{d\varepsilon} = -[D_{\varphi\varphi}^2 \Phi]^{-1} \frac{\partial}{\partial\varepsilon} \nabla_\varphi \Phi$, the authors capture the evolution of the optimal solution $\gamma(\varepsilon)$ along the entire regularization path, enabling simultaneous computation of the full curve and derivatives at $\varepsilon=0$. They prove well-posedness of the ODE and provide a closed-form second-derivative formula $C''(0)$ for the unconstrained two-marginal case, with higher derivatives accessible through the same framework. Numerical experiments across two-, three-, and multi-period martingale OT, as well as geodesics and barycenters, confirm that the ODE method matches Sinkhorn results and offers computational advantages by yielding the entire solution path, including entropic interpolants.

Abstract

We characterize the solution to the entropically regularized optimal transport problem by a well-posed ordinary differential equation (ODE). Our approach works for discrete marginals and general cost functions, and in addition to two marginal problems, applies to multi-marginal problems and those with additional linear constraints. Solving the ODE gives a new numerical method to solve the optimal transport problem, which has the advantage of yielding the solution for all intermediate values of the ODE parameter (which is equivalent to the usual regularization parameter). We illustrate this method with several numerical simulations. The formulation of the ODE also allows one to compute derivatives of the optimal cost when the ODE parameter is $0$, corresponding to the fully regularized limit problem in which only the entropy is minimized.

Figures (6)

• Figure 1: Evolution of the primal coupling at 16 evenly distributed values of $\varepsilon$ between 0 and 1 calculated by the ODE method for two marginal optimal transport with attractive cost $c(x, y) = |y - x|^2$.
• Figure 2: Evolution of the primal coupling at 16 evenly distributed values of $\varepsilon$ between 0 and 1 calculated by the ODE method for two marginal optimal transport with repulsive cost $c(x, y) = -\log(0.1 + |x - y|)$.
• Figure 3: The optimal measure for three marginal optimal transport with cost $c(x, y, z) = d(x,y) + d(y, z) + d(x, z)$ where $d(x,y) = -\log(0.1 + |x - y|)$ generated by ODE
• Figure 4: The optimal measure for martingale optimal transport with cost $c(x, y) = \exp(-x) \cdot y^2$ generated by ODE
• Figure 5: 2D projections for pairs of marginals of the optimal measures for multi-period martingale transport problem. Top: generated by ODE method. Bottom: generated by Sinkhorn algorithm

Theorems & Definitions (20)

• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Proposition 3.4