Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport

2026-06-03

AI Summary

This work addresses the challenge of high-dimensional optimal transport, which generally lacks closed-form solutions and struggles to preserve the geometric structure and directional information of the original space. The authors introduce cone-compatible Monge geometry, leveraging a partial order induced by convex cones to extend the one-dimensional monotone rearrangement to higher dimensions. By integrating squared Mahalanobis costs, cone duality theory, and Monge’s exchange inequality, they establish sharp characterizations of cone compatibility, enabling, for the first time, closed-form optimal couplings in high-dimensional ordered data under the original metric. The proposed framework offers interpretability and directional fidelity while guaranteeing feasibility, duality, stability, approximation bounds, Gaussian recovery, and favorable statistical properties. Additionally, it defines a cone-chain Wasserstein distance and provides an efficient computational scheme.

Abstract

High-dimensional optimal transport is seldom available in closed form. The one-dimensional case is exceptional because the order of the real line is compatible with convex transport costs, making monotone rearrangement optimal. This paper studies when an analogous Monge structure can be recovered in higher dimensions from a partial order. We introduce a cone-compatible Monge geometry: a closed convex cone (K) induces the order (x\preceq_K y) whenever (y-x\in K), and is compatible with a cost if ordered pairs satisfy a Monge exchange inequality. For squared Mahalanobis costs (c_M(x,y)=(x-y)^\top M(x-y)), we prove a sharp characterization: compatibility holds exactly when (K) is acute under the (M)-inner product, namely (u^\top Mv\ge0) for all (u,v\in K), equivalently (K\subseteq K_M^*). Under this condition, measures supported on cone chains admit a quantile-type closed-form optimal coupling, yielding exact transport under the original ground cost rather than after projection or metric replacement. We distinguish the resulting cone-chain Wasserstein metric on canonically ordered chain distributions from an extended directed cone transport cost on general measures, and develop feasibility, duality, stability, approximation, Gaussian recovery, statistical, and computational results. The theory is complementary to sliced and tree Wasserstein distances: it is not a universal fast surrogate, but a way to obtain interpretable, direction-valid, original-space monotone transport for ordered high-dimensional data.

Problem

• optimal transport
• Monge geometry
• high-dimensional
• partial order
• cone compatibility

Innovation

• cone-compatible Monge geometry
• optimal transport
• partial order
• Mahalanobis cost
• cone chains