Minimum Volume Optimal Transport for Multivariate Conformal Prediction
Abstract
Optimal transport (OT) enables multivariate conformal prediction (CP) to construct flexible prediction sets that reflect the underlying complex distributional geometry while retaining distribution-free coverage guarantees. Existing OT-based methods typically rely on a fixed Euclidean transport cost, which can be suboptimal in terms of the volume of prediction sets. We propose Minimum-Volume OTCP, a framework that learns a Mahalanobis ground cost to directly minimize prediction-region volume over a class of positive-definite metrics. Using a metric Brenier representation, we derive a volume-based objective that depends explicitly on the Mahalanobis cost matrix and optimize it jointly with the transport map. We establish finite-sample marginal coverage of the output prediction sets and optimality of volume metrics. We further develop an individualized extension that learns covariate-dependent transport metrics, adapting the shape and orientation of prediction sets to heterogeneous conditional distributions. For this extension, we also show the asymptotic conditional coverage of our method. Simulations and real-data experiments demonstrate substantial reductions in prediction-set volume.
Related
- Minimum Volume Conformal Sets for Multivariate Regression
- Generalized Hyperrectangular Prediction Sets in Multivariate Conformal Prediction
- Tractable Relaxations of Multivariate Stochastic Dominance via Optimal Transport and CVaR
- Finite-Sample Optimal Conformal Prediction Sets under Covariate Shift
- Multivariate conformal uncertainty propagation in multitask atomistic simulation: Successes and pitfalls
BibTeX
@misc{zhang2026minimum,
title = {{Minimum Volume Optimal Transport for Multivariate Conformal Prediction}},
author = {Chuchen Zhang and Haojie Ren and Zhaojun Wang and Yajie Bao and Changliang Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14907},
eprint = {2610.14907},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14907},
}
RIS
TY - UNPB
TI - Minimum Volume Optimal Transport for Multivariate Conformal Prediction
AU - Chuchen Zhang
AU - Haojie Ren
AU - Zhaojun Wang
AU - Yajie Bao
AU - Changliang Zou
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14907
UR - https://arxiv.org/abs/2610.14907
AB - Optimal transport (OT) enables multivariate conformal prediction (CP) to construct flexible prediction sets that reflect the underlying complex distributional geometry while retaining distribution-free coverage guarantees. Existing OT-based methods typically rely on a fixed Euclidean transport cost, which can be suboptimal in terms of the volume of prediction sets. We propose Minimum-Volume OTCP, a framework that learns a Mahalanobis ground cost to directly minimize prediction-region volume over a class of positive-definite metrics. Using a metric Brenier representation, we derive a volume-based objective that depends explicitly on the Mahalanobis cost matrix and optimize it jointly with the transport map. We establish finite-sample marginal coverage of the output prediction sets and optimality of volume metrics. We further develop an individualized extension that learns covariate-dependent transport metrics, adapting the shape and orientation of prediction sets to heterogeneous conditional distributions. For this extension, we also show the asymptotic conditional coverage of our method. Simulations and real-data experiments demonstrate substantial reductions in prediction-set volume.
ER -