Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method

TL;DR

This work develops a Newton-based solver for semi-discrete optimal transport with costs $c(x,y)=\sum_{l=1}^m \alpha_l \|x-y\|_{p_l}$, $1<p_l<\infty$, establishing that Laguerre cells are star-shaped with respect to target points and leveraging this in an efficient 2D boundary-tracking framework. The method hinges on a convex potential $\Phi(w)$ with gradient $\nabla\Phi(w)=\big(\mu(A(y_i)) - \nu_i\big)$ and a Hessian expressed as 1D boundary integrals, enabling precise Newton updates after projecting away the Hessian's 1D nullspace. An adaptive, error-controlled algorithm combines boundary tracing (shooting, predictor-corrector, breakpoint handling) with adaptive 1D integration to compute both gradients and Hessians, while initial guesses are obtained via grid-based or homotopy strategies and damped Newton steps ensure feasibility. Extensive 2D computational experiments show high accuracy and performance gains over competing approaches, highlighting the practical viability of Newton-based semi-discrete OT for a broad class of $p$-norm costs. The work also introduces a conditioning metric $\kappa(w)$ to quantify problem well-posedness and discuss implications for robustness and extensions to higher dimensions and less smooth densities.

Abstract

In this work, we propose a novel implementation of Newton's method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of $p$-norms, $1<p<\infty$. It is well understood that the solution of a semi-discrete OT problem is equivalent to finding a partition of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton's method to find the sought network structure. We provide implementation details and extensive results in support of our technique in 2-d problems, as well as comparison with other approaches used in the literature.

Figures (10)

• Figure 1: For the same target points, $w$, and cost: (A) Laguerre cells in ${\mathbb R}^2$ and (B) Restricted to the unit square.
• Figure 2: Counterexamples to uniqueness $w=0$ The shaded regions are in both $A(y_1)$ and $A(y_2)$.
• Figure 3: Different $p$-norm costs
• Figure 4: $p$-norm costs for growing $p$
• Figure 5: $p$-norm costs for $p$ decreasing toward $1$

Theorems & Definitions (71)

• Definition 2.5: Laguerre tessellation
• Remark 2.6
• Lemma 2.7: Every $x\in {\mathbb R}^n$ is in some ${L}(y_i)$
• proof
• Definition 2.8: Active boundary and interior
• Lemma 2.9: Existence of boundary between cells: necessary condition
• proof
• Lemma 2.10: $y_i\in {L}(y_i)$
• proof
• Lemma 2.11