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

Abstract

Closed-form solutions for high-dimensional optimal transport are rarely attainable. The one-dimensional scenario stands out as an exception, where the natural ordering of the real line aligns with convex transport costs, thereby rendering monotone rearrangement the optimal strategy. This study investigates the conditions under which a similar Monge structure can be reconstructed in higher dimensions using a partial order. We propose a framework termed cone-compatible Monge geometry, wherein a closed convex cone ($K$) establishes an order ($x\preceq_K y$) defined by $y-x\in K$. This geometry is considered compatible with a specific cost function if ordered pairs adhere to a Monge exchange inequality.

For squared Mahalanobis costs, defined as $c_M(x,y)=(x-y)^\top M(x-y)$, we establish a precise characterization: compatibility is achieved if and only if the cone $K$ is acute with respect to the $M$-inner product. This condition requires that $u^\top Mv\ge0$ for all vectors $u,v\in K$, which is equivalently expressed as $K\subseteq K_M^*$. When this criterion is met, measures supported on cone chains possess a quantile-type closed-form optimal coupling. This approach facilitates exact transport under the original ground cost, avoiding the need for projection or metric substitution.

Furthermore, we differentiate between the cone-chain Wasserstein metric applied to canonically ordered chain distributions and an extended directed cone transport cost utilized for general measures. The paper presents comprehensive findings regarding feasibility, duality, stability, approximation, Gaussian recovery, statistical properties, and computational aspects. Rather than serving as a universal, rapid surrogate like sliced or tree Wasserstein distances, this theory offers a method to achieve interpretable, directionally valid, and monotone transport within the original space for high-dimensional data that possesses an inherent order.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC