Curvature bounds for the cost of coherence in conditional optimal transport
Abstract
We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order r4 as r→0 on squares of half-side length r with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.
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BibTeX
@misc{wang2026curvature,
title = {{Curvature bounds for the cost of coherence in conditional optimal transport}},
author = {Xiaozhen Wang and Xiang Zhou and Zihan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09738},
eprint = {2610.09738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09738},
}
RIS
TY - UNPB
TI - Curvature bounds for the cost of coherence in conditional optimal transport
AU - Xiaozhen Wang
AU - Xiang Zhou
AU - Zihan Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09738
UR - https://arxiv.org/abs/2610.09738
AB - We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order r^4 as r→0 on squares of half-side length r with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.
ER -