Multivariate Conformal Prediction Using Optimal Transport

Abstract

Conformal prediction (CP) provides distribution-free uncertainty quantification by constructing prediction sets whose validity relies on ranking conformity scores. Because ranking requires an ordering, most CP methods use univariate scores; extending them to multivariate settings, where no canonical order for vectors exists, remains challenging. We build on the theory of Monge--Kantorovich quantiles and ranks to propose a geometry-aware scalarization of vector-valued scores: we transport multivariate conformity scores to the spherical uniform distribution on the unit ball via an entropic optimal transport (OT) map and use the transported radius as a scalar score. Standard split conformal calibration then applies directly, preserving finite-sample marginal coverage. The resulting method, OTCP, produces prediction regions that adapt to the empirical geometry of the score distribution, going beyond the ellipsoidal sets imposed by norm-based scalarizations. Across a benchmark of 24 multivariate regression datasets, OTCP improves efficiency and conditional-coverage metrics mainly in low output dimensions ($d \leq 4$), while we also study the computational and statistical trade-offs involved in estimating entropic OT maps.