# From Cone Geometry to Monge Structure: Local Defects and Global Regret in High-Dimensional Optimal Transport

arXiv: https://arxiv.org/abs/2606.04695 · HTML: https://mlsanity.com/paper/2606.04695/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 50 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 22 of 22)

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- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Morpho-MNIST: Quantitative Assessment and Diagnostics for Representation Learning](https://mlsanity.com/paper/1809.10780/) (2018)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Causal transport in discrete time and applications](https://mlsanity.com/paper/1606.04062/) (2016)
- [Existence, Duality, and Cyclical monotonicity for weak transport costs](https://mlsanity.com/paper/1809.05893/) (2018)
- [FlashSinkhorn: IO-Aware Entropic Optimal Transport on GPU](https://mlsanity.com/paper/2602.03067/) (2026)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [Estimation of Monge Matrices](https://mlsanity.com/paper/1904.03136/) (2019)
- [Order Constraints in Optimal Transport](https://mlsanity.com/paper/2110.07275/) (2021)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025)
- [The Directional Optimal Transport](https://mlsanity.com/paper/2002.08717/) (2020)
- [Revisiting transportation problems under Monge costs with applications to location problems](https://mlsanity.com/paper/2602.15151/) (2026)
- [Active Seriation: Efficient Ordering Recovery with Statistical Guarantees](https://mlsanity.com/paper/2603.15336/) (2026)

## Abstract (arXiv)

Exact optimal transport in high dimensions becomes tractable only when additional structure selects the optimal coupling. Classical Monge transportation solves the discrete problem once a cost array is known to be Monge, but it does not explain when geometry in the original feature space creates that structure. We characterize this regime for cone-induced orders and squared Mahalanobis costs: Macuteness of the cone is equivalent to the Monge property of every cross-cost matrix generated by two cone chains, after which classical northwest-corner optimality gives exact transport. When the inequalities fail, we derive an exact variational representation of northwestcorner regret over a marginal-dependent cumulative-deficit polytope. Its Fr'echet box relaxation yields a projection-free, marginal-weighted localdefect bound; we characterize when the relaxation is exact and prove that it is worst-case sharp when only local defect budgets are known. We further show that the worst-case defect patterns can be realized by genuine squared-Euclidean cross-costs. Finally, Mahalanobis distances of observed increments to the cone control the local defects, while a strict geometric margin gives an explicit radius of exact stability. The resulting theory links high-dimensional ground-space geometry to exact Monge transport and quantifies the global loss induced by local departures from ordered structure.