Optimal Transport Is Not Volume-Optimal for Conformal Prediction

Transport-based methods construct multivariate prediction regions as pullbacks of quantile regions of a simple reference distribution: any invertible map that transports the reference measure to the conditional law of the response yields regions with exact conditional coverage. This validity, however, says nothing about the volume of the resulting regions. Recent work has favored optimal-transport (Brenier) maps, implicitly treating the induced regions as canonical. We show that this choice is provably suboptimal. For a standard Gaussian reference, we show that, among elliptical targets with finite second moments, exactly the Gaussian and spherical targets yield prediction regions whose volumes equal those of the corresponding highest-density regions (HDR) at every coverage level; for every other target in this class, the Brenier-induced region has strictly greater Lebesgue volume over a nonempty open interval of coverage levels. For the compactly supported references used in center-outward multivariate ranks — the uniform ball and the spherical uniform — the same volume gap holds for every non-spherical elliptical target, without exception. We then factor admissible transports into a Brenier map and a measure-preserving rearrangement, thereby separating coverage from volume. We parameterize this class using smooth measure-preserving flows generated by skew-symmetric matrix fields with at most \(2(d_y-1)\) nonzero entries and derive a stochastic volume objective. Experiments on synthetic and real data show that measure-preserving rearrangements yield smaller prediction regions.