Characterizing and computing solutions to regularized semi-discrete optimal transport via an ordinary differential equation

TL;DR

The paper tackles the semi-discrete OT problem with entropic regularization for general costs by deriving a dual formulation $\Phi(\psi,t)$ and proving that the dual potentials along the regularization path $t=1-\varepsilon$ satisfy a well-posed ODE: $\nabla^2_{\psi,\psi}\Phi(\psi(t),t)\,\psi'(t)+\frac{\partial}{\partial t}\nabla_\psi \Phi(\psi(t),t)=0$. It establishes uniform Hessian convexity bounds on appropriate regimes, ensures existence and uniqueness of solutions to the ODE (including extension to $t=1$ under twisted/generic conditions), and shows how to recover the unregularized OT from the ODE limit. Numerically, the ODE-based solver, implemented via a third-order Runge–Kutta method, is competitive with Newton's method for squared Euclidean costs and superior for other costs or when target points lie outside the source domain. The framework also provides convergence-rate insights as the regularization vanishes and offers a robust approach to compute both entropically regularized and unregularized semi-discrete OT in higher dimensions. Overall, this work links convex analysis, Laguerre-cell geometry, and ODE theory to yield a principled, scalable algorithm for semi-discrete OT with general costs.

Abstract

This paper investigates the semi-discrete optimal transport (OT) problem with entropic regularization. We characterize the solution using a governing, well-posed ordinary differential equation (ODE). This naturally yields an algorithm to solve the problem numerically, which we prove has desirable properties, notably including global strong convexity of a value function whose Hessian must be inverted in the numerical scheme. Extensive numerical experiments are conducted to validate our approach. We compare the solutions obtained using the ODE method with those derived from Newton's method. Our results demonstrate that the proposed algorithm is competitive for problems involving the squared Euclidean distance and exhibits superior performance when applied to various powers of the Euclidean distance. Finally, we note that the ODE approach yields an estimate on the rate of convergence of the solution as the regularization parameter vanishes, for a generic cost function.

Figures (8)

• Figure 1: Mass of Laguerre cells in 2-d as a function of $t$
• Figure 2: Time evolution of Laguerre cells
• Figure 3: Time Evolution of Laguerre cells in Example \ref{['E4']} with $b=0.5$
• Figure 4: Time evolution of measures
• Figure 5: Time evolution of Laguerre cells with 10 random points

Theorems & Definitions (28)

• Remark 2.1
• Definition 2.2: Laguerre cell
• Definition 2.3: Smoothed Laguerre cell
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 2.7: Strong convexity of $\Phi$
• proof