Minimum Volume Optimal Transport for Multivariate Conformal Prediction

Chuchen Zhang; Haojie Ren; Zhaojun Wang; Yajie Bao; Changliang Zou

stat.MLOct 10, 2026

Minimum Volume Optimal Transport for Multivariate Conformal Prediction

Authors: Chuchen Zhang, Haojie Ren, Zhaojun Wang, Yajie Bao, Changliang Zou

Organizations: School of Statistics and Data Science, Nankai University · School of Mathematical Sciences, Shanghai Jiao Tong University · School of Management, Fudan University

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.

Figures & tables

Figure 1: Metric choices on an elliptical score distribution. The accepted assignment regions are finite unions of polyhedral cells (Appendix B ) under

M=I

M\propto\Sigma^{-1/2}

, and

M\propto\Sigma^{-1}

. Among the three choices,

M\propto\Sigma^{-1/2}

yields the smallest symmetric difference from the oracle region.

Figure 2: Homogeneous settings. Dashed lines mark nominal coverage and setting-specific oracle minimum volumes.

Figure 3: Aspect-ratio ablation for Five-star and Clover settings. Translucent bars report the mean paired improvement of MV-OTCP over FOTCP on the right axis. The dashed line in the marginal-coverage panels marks the nominal target.

Figure 4: Heterogeneous settings. Boxplots show marginal and worst-case coverage; bars show root MSCE and set volume. Dashed lines mark nominal coverage.

Figure 5: Localized ground-cost and anchor ablation. Translucent bars report the mean paired volume improvement of learned-metric IMV-OTCP over IMV-OTCP(

M=I

Metric	Dataset	SCP	MVCS	RLCP	IMV-OTCP
Marginal coverage	Energy	$0.909(0.004)$	$0.910(0.004)$	$0.912(0.003)$	$0.917(0.004)$
	Bias	$0.901(0.002)$	$0.901(0.002)$	$0.903(0.002)$	$0.901(0.002)$
	Taxi	$0.901(0.001)$	$0.902(0.001)$	$0.903(0.001)$	$0.900(0.001)$
	rf1	$0.900(0.001)$	$0.901(0.001)$	$0.901(0.001)$	$0.900(0.002)$
	scm1d	$0.898(0.001)$	$0.899(0.001)$	$0.901(0.001)$	$0.897(0.002)$
$\sqrt{\mathrm{MSCE}}$	Energy	$0.115(0.0039)$	$0.101(0.0028)$	$0.104(0.0032)$	$0.091(0.0022)$

Table 1: Regression benchmarks over

100

splits. Entries are means (standard errors). Root volume is the median of

V_{i}^{1/{d}}

. Bold marks the lowest root volume in each dataset.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: General task-objective experiments. The rows correspond to the integrated and smoothed quadratic settings. The dashed line in the marginal-coverage panels marks the nominal target; lower task objective and volume are preferable.

Figure 7: U.S. equity portfolio results for ellipsoidal uncertainty sets. The dashed line is the nominal level

0.9

. Certificate and decision loss are means with normal-approximation

95\%

intervals (mean

\pm 1.96

standard errors). Lower is better for the certificate and decision loss.

Dataset	$n$	$d_{\mathcal{X}}$	${d}$	$n_{\rm tr}$	$n_{\rm est}$	$n_{\rm cal}$	$n_{\rm test}$
Energy	768	8	2	422	115	115	116
Bias	7,752	22	2	5,426	775	775	776
Taxi	61,286	6	2	42,900	1,000	1,000	6,130
rf1	9,125	64	8	6,387	912	912	914
scm1d	9,803	280	16	6,862	980	980	981

Appendix

Table 2: Real multivariate-regression datasets and effective split sizes. Estimation and calibration are capped at

1{,}000

observations; the test split is not truncated.

Training	Estimation	Calibration	Test
2012–2014	2015	2016	2017

Appendix

Table 3: Leakage-free chronological split for the portfolio experiment.

Conformal Prediction

Conformal prediction wraps predictive models with calibrated sets or intervals that provide statistical coverage guarantees.

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Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings

Eugene Ndiaye

Apple

Uncertainty Quantification Conformal Prediction Optimal Transport Set-Valued Prediction

Open paper AskPDF arXiv

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in

O(n^3)

time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

Abstract

Conformal Prediction

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