Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport
Mirrored from arXiv — Machine Learning for archival readability. Support the source by reading on the original site.
Computer Science > Machine Learning
Title:Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport
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.
| Comments: | 13 pages, 2 figures, including appendices |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2606.04695 [cs.LG] |
| (or arXiv:2606.04695v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.04695
arXiv-issued DOI via DataCite (pending registration)
|
Access Paper:
- View PDF
- HTML (experimental)
- TeX Source
References & Citations
Bookmark
Bibliographic and Citation Tools
Code, Data and Media Associated with this Article
Demos
Recommenders and Search Tools
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
More from arXiv — Machine Learning
-
Early Detection of Alzheimer's Disease Using Explainable Machine Learning on Clinical Biomarkers: A Multi-Class Classification Study Using the Alzheimer's Disease Neuroimaging Initiative (ADNI) Dataset
Jun 4
-
Novel Aspects of IEEE SA P3109 Arithmetic Formats for Machine Learning
Jun 4
-
Position: Deployed Reinforcement Learning should be Continual
Jun 4
-
Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent
Jun 4
Discussion (0)
Sign in to join the discussion. Free account, 30 seconds — email code or GitHub.
Sign in →No comments yet. Sign in and be the first to say something.