Conditional Random Ordered Transport Spaces

2026-06-06

AI Summary

This work addresses the inability of the classical Wasserstein distance to capture directional structure in probability distribution transformations under semantic, causal, physical, or risk-sensitive prior constraints. To overcome this limitation, the authors propose the Conditional Random Ordered Transport Space (CROTS) framework, which integrates stochastic partial orders, conditional risk functionals, and Wasserstein geometry to define admissible transport structures that support both hard and soft ordered mass displacements. This study is the first to unify stochastic ordering, conditional information, and optimal transport, establishing the well-posedness, duality, and completeness of CROTS. It further reveals the independent recursive behavior of local order violations under Wasserstein convergence and derives an asymptotic lower bound on order-preserving risk, providing a rigorous mathematical characterization of distributional shift and robustness failure.

Abstract

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 \(L^0\)-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.

Problem

• ordered transport
• Wasserstein distance
• admissible transformation
• conditional risk
• distributional learning

Innovation

• Conditional Random Ordered Transport Spaces
• Ordered Optimal Transport
• Stochastic Order
• Wasserstein Metric
• Conditional Risk Functional