# Conditional Random Ordered Transport Spaces

arXiv: https://arxiv.org/abs/2606.08113 · HTML: https://mlsanity.com/paper/2606.08113/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 31 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds](https://mlsanity.com/paper/2403.06560/) (2024)
- [Nuclear Norm based Matrix Regression with Applications to Face Recognition with Occlusion and Illumination Changes](https://mlsanity.com/paper/1405.1207/) (2014)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [ERDDCI: Exact Reversible Diffusion via Dual-Chain Inversion for High-Quality Image Editing](https://mlsanity.com/paper/2410.14247/) (2024)
- [Straighten Viscous Rectified Flow via Noise Optimization](https://mlsanity.com/paper/2507.10218/) (2025)
- [Stochastic order on metric spaces and the ordered Kantorovich monad](https://mlsanity.com/paper/1808.09898/) (2018)
- [Towards Better Spherical Sliced-Wasserstein Distance Learning with Data-Adaptive Discriminative Projection Direction](https://mlsanity.com/paper/2412.19212/) (2024)
- [L2 over Wasserstein: Statistical Analysis for Optimal Transport](https://mlsanity.com/paper/2605.21365/) (2026)

## Abstract (arXiv)

A small Wasserstein distance does not certify that a transformation is admissible. In evidence-constrained, semantic, causal, physical, monotone, or risk-sensitive learning, one must ask not only how far two probability laws are, but whether mass has moved in a direction allowed by available information. We introduce conditional random ordered transport spaces (CROTS), a class of (L0)-valued spaces of random probability measures equipped with a Wasserstein ambient metric, a closed stochastic order, hard and soft ordered transport discrepancies, and a conditional risk functional for evaluating order violation under an evidence sigma-field. The central object is an order-admissible transport geometry for random measure-valued dynamics, distinct from cone-valued metrics, ordered Kantorovich constructions, random Wasserstein spaces alone, and model-specific residuals for generative paths. We develop the foundations of CROTS as a space theory for reliable distributional learning. The results include well-posedness and duality for hard and soft ordered transport, soft-to-hard variational convergence, measurability and completeness of the random lifted space, reductions to classical Wasserstein and ordered geometries, ordered geodesics, constrained barycenters and projections, conditional risk-transport duality, and separation of order-violating distributions. The main stability theorem shows that random learning dynamics may converge in the ambient Wasserstein metric while its local admissibility leakage follows a separate conditional order-risk recursion. The resulting asymptotic order-risk floor provides a mathematical language for evidence overreach, ordered distribution shift, robustness failure, and admissible distributional dynamics.