Multivariate Conformal Prediction using Optimal Transport

Eugene Ndiaye; Louis Bethune; Marco Cuturi; Michal Klein

REVIEW 3 major objections 5 minor 1 cited by

Multivariate Conformal Prediction using Optimal Transport

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read By mapping vector-valued conformity scores through an optimal transport map to a uniform ball and calibrating a radius on the resulting norms, this paper constructs conformal prediction sets for multivariate outputs with a…

desk verdict A promising OT-based multivariate conformal method whose headline coverage guarantee does not cover the implementation actually used in the experiments. read the letter →

arxiv 2502.03609 v1 pith:6PQCYIWI submitted 2025-02-05 stat.ML cs.LG

Michal Klein , Louis Bethune , Eugene Ndiaye , Marco Cuturi This is my paper · ORCID

classification stat.MLcs.LG MSC 62G1562H05

keywords conformalpredictionoptimaltransportmultivariateregressiondistribution-freecoverageentropicmapconformityscorequantileregionuncertaintyquantification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Conformal prediction builds uncertainty sets by ranking a scalar score, but vector-valued outputs have no canonical ordering. This paper proposes OT-CP, which transports vector-valued conformity scores to a uniform ball in $\mathbb{R}^d$ via an optimal transport map and uses the norm of the transported score as a univariate rank, so that ordinary conformal calibration applies. The paper's central theoretical claim is a distribution-free finite-sample coverage guarantee that holds for any approximation of the transport map, because the calibration radius is an empirical quantile of the transported norms rather than a quantity tied to map accuracy. On a benchmark of multivariate regression tasks, OT-CP produces smaller mean region sizes than box-based and norm-based baselines on most low-dimensional datasets, at higher computational cost.

What carries the argument

The central object is the entropic Brenier map estimator $T_\varepsilon(z) = z - \nabla f_\varepsilon(z)$, built from Sinkhorn dual potentials, which gives a tractable, out-of-sample evaluable map from residual space to a stratified spherical uniform grid. The norm of its image is the scalar conformity score, and the discrete spherical uniform target—$n_S$ directions times $n_R$ radii plus $n_o$ origin copies—yields the exact radius formula in Proposition 3.2. The mechanism that carries the argument is the empirical quantile of transported norms, which absorbs any approximation error in the map into the calibration radius.

What would settle it

Take an i.i.d. multivariate regression dataset, set $\alpha=0.1$, fit the entropic map on the calibration residuals, compute the radius as the empirical quantile of the mapped calibration norms, score a fresh test point with the same fitted map, and repeat over many test points: if coverage ever falls systematically below 90 percent, the guarantee as stated is false for that protocol. A sharper test is the transductive version: refit the map on all $n+1$ points in each trial using an order-dependent optimization and then check marginal coverage, since exchangeability of the transformed scores is the only assumption the proof needs.

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Extended reading notes

Core claim

The central claim is that multivariate conformal prediction reduces to a univariate conformal problem after a change of coordinates learned by optimal transport. Given vector-valued residuals $S(X_i,Y_i)$, the method learns a map $\hat T$ pushing their empirical distribution toward a uniform measure on the unit ball, defines the score of any candidate $y$ as $\|\hat T(S(X_{n+1}, y))\|$, and sets the prediction region as the preimage of a ball in transported space whose radius is the empirical $(1-\alpha)$-quantile of the observed transported norms. Proposition 3.6 asserts that this set contains $Y_{n+1}$ with probability at least $1-\alpha$ for any finite sample and any map $\hat T$, as long as the sequence of transformed scores is exchangeable. Because the radius is calibrated on the actual distribution of $\|\hat T(S)\|$, approximation error in the transport map affects only the size and shape of the region, not the coverage guarantee.

Load-bearing premise

The load-bearing premise is that the transformed scores $\|\hat T(S_i)\|$ for calibration and for the new point are exchangeable with one another, which requires the transport map to be fixed before calibration or to be a symmetric function of the whole exchangeable sequence; if the map is refit using the same data that determines the radius in a way that breaks that symmetry, the coverage guarantee can fail.

Editorial extensions

If this is right

Calibration on the empirical quantile of transported norms absorbs transport-map approximation error, so coverage stays valid even with entropic regularization and finite target grids.
OT-CP regions are preimages of balls in transported space, so they can be non-elliptic and shaped by the residual distribution, unlike ellipsoidal Merge-CP sets.
When source and target distributions are Gaussian or uniform, the transport map is affine and OT-CP's score coincides with the Mahalanobis distance, making the method a strict generalization of Merge-CP with a Mahalanobis norm.
On the 24-task benchmark, OT-CP reports smaller mean region sizes than the M-CP and Merge-CP baselines on most low-dimensional datasets, with the advantage shrinking for outputs of dimension 14 to 16.
The computational cost is $O(nm)$ to fit the map and $O(m)$ per candidate score, so the number $m$ of target grid points and the regularization $\varepsilon$ trade region size against runtime, with $m=2^{15}$ and $\varepsilon=0.1$ working well in the experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the coverage guarantee is independent of map quality, the honest comparison metric for OT-CP becomes region size alone: a poor transport map still gives valid sets, just larger ones, so the method's practical value hinges on how well entropic maps capture residual geometry in modest dimensions.
The scalar calibration step means OT-CP can inherit univariate conformal extensions, such as covariate-shift weighting, adaptive or conditional quantile calibration, or online exchangeability adjustments, by swapping the final ranking step without touching the transport machinery.
A practitioner who reuses the calibration set to tune $\varepsilon$ or $m$ and then reports coverage on the same set is outside the exchangeability premise; a clean protocol is to choose hyperparameters on training data and reserve the calibration split for the radius.
The framework suggests replacing the spherical-uniform target with a data-dependent or density-shaped target, which could trade away uniform-norm calibration simplicity for smaller regions at the cost of a new calibration rule.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the idea is genuinely good: use OT ranks to turn a multivariate score into a scalar, then apply conformal calibration to that scalar. That's a natural combination, and the entropic-map implementation makes it tractable. Second, the headline coverage guarantee (Prop 3.6) doesn't apply to the algorithm they actually run. That's the load-bearing issue.

Here's the problem. Prop 3.6 and Lemma B.2 prove coverage for a map T̂_{n+1} that is a permutation-invariant function of all n+1 scores—the transductive/full-conformal version. The implementation in Section 3.4 instead fits the entropic map once on the n calibration residuals, then computes the empirical quantile of the norms of those same residuals, and uses that as the radius for a test point transported by the same map. Those transformed calibration norms aren't exchangeable with the test norm, because the map has already seen the calibration points. The symmetry argument in Lemma B.2 fails, and no finite-sample coverage guarantee covers that protocol. Remark 3.3 even acknowledges that coverage doesn't hold if the map is computed on a hold-out independent dataset, but that's not what they mean—they mean the discrete assignment version doesn't transfer. The paper doesn't crisply separate the two protocols, and the abstract claims distribution-free coverage for the method as implemented.

The fix is simple in principle: fit the entropic map on a separate training set of residuals, or at least on a different split, and then calibrate on a fresh hold-out. Standard split conformal then applies to the fixed map. As written, the main theorem proves a different method than the one in the experiments.

Softer issues: Definition 2.3 says ∥T(Z)∥ is uniform on (0,1) for the uniform distribution on the unit ball; that's only true for a nonstandard radial-uniform target. This doesn't break the conformal calibration (which uses empirical quantiles), but it's a mathematical error in the exposition. The hyperparameters ε and m are tuned on the same benchmark, which inflates the reported gains; the high-dimensional results collapse anyway, which the authors acknowledge. The low-dimensional gains are real, as far as I can tell.

Who should read it: people working on multivariate conformal regression or OT-based ranks. It deserves peer review—the combination is novel and the experimental comparison is extensive—but the referee should insist that the guarantee match the implementation, and that the hyperparameter selection be out-of-sample.

Referee Report

3 major / 5 minor

Summary. The paper proposes OT-CP, a conformal prediction method for multivariate outputs in which vector-valued conformity scores are mapped to a univariate rank via an optimal transport map to a spherical uniform target, after which standard univariate conformal prediction is applied to the resulting norms. The authors claim a distribution-free finite-sample coverage guarantee for the resulting region (Proposition 3.6, Lemma B.2) and report smaller mean region sizes than M-CP and Merge-CP baselines on most low-dimensional benchmark datasets, with ablations over the entropic regularization ε and the number of target points m.

Significance. The conceptual idea of using optimal transport to define a multivariate depth or rank and then inheriting conformal guarantees is attractive and timely, and the paper includes a broad empirical study (24 datasets, ablations, runtime comparisons). The theoretical core, however, is a restatement of the standard exchangeability argument for conformal prediction; the paper's distinctive claim is that the guarantee holds for any approximated transport map, and that claim is where the proof and the implementation diverge. If the protocol issues are resolved, the method could be a useful addition to the multivariate conformal toolbox; in its current form, the finite-sample coverage guarantee is not established for the implemented algorithm.

major comments (3)

[§3.3 / Lemma B.2 / §3.4] The coverage guarantee in Proposition 3.6 and Lemma B.2 requires the empirical quantile region bR_{α,n+1} to be a symmetric function of the full exchangeable sequence (Z1,...,Z_{n+1}). This holds only when the transport map is either a fixed map independent of the calibration data (split conformal) or a permutation-invariant function of all n+1 points (transductive). The implementation in Section 3.4 fits the entropic map Tε on the n calibration residuals and then calibrates the radius r̂ as the empirical quantile of the norms of those same residuals; the test score is transported by a map that was not fitted on it, so the transformed norms are not exchangeable. The averaging identity in Lemma B.2 therefore fails for the reported algorithm, and Proposition 3.6 does not cover the implementation. Remark 3.3's statement that the coverage guarantee 'does not hold if the transport map is computed solely on a hold-out independent dataset' conflates the calibration set with the split; the fix is to train the map on a separate training split (or use a pre-trained map) and calibrate on a disjoint set, or to recompute the map for each candidate transductively.
[Definition 2.3 and Eq. (7), Eq. (14)] Definition 2.3 and Eq. (7): the claim that ||T(Z)|| is uniform on (0,1) when T pushes P to the 'uniform distribution on the unit ball' is only correct for the radial-uniform (spherically uniform) target measure, not for the Lebesgue-uniform measure on the ball, for which ||U|| has CDF r^d. The paper's own discrete construction in Section 3.1 is radial-uniform, so Definition 2.3 and Eq. (7) are internally inconsistent. Similarly, Eq. (14) states that ||T_{n+1}(Z_{n+1})|| follows U_{n+1}({0,1/nR,...,1}), but with nS directions per shell and equal mass per grid point, the radius distribution assigns mass nS/(n+1) to each positive shell (plus no/(n+1) at the origin), which is not uniform over the listed atoms. The definitions should be corrected to use the spherical uniform measure explicitly and the displayed distributional statements should be fixed accordingly.
[Section 3.3, Eq. (19)] Proposition 3.6 is stated for an arbitrary approximation map T̂ with no condition, while the proof in Lemma B.2 requires the symmetry of bR_{α,n+1} in the full data vector. This is a nontrivial condition that fails for the entropic-map implementation described in Section 3.4. The proposition should state explicitly whether T̂ is fixed (split conformal) or permutation-invariant in the full data vector, and the claim that the guarantee holds 'for any approximation map' should be withdrawn or appropriately restricted.

minor comments (5)

[Section 3.2] The phrase 'any approximation ˆT that preserves the permutation invariance of the score function' is undefined; a fixed pre-trained map is not a function of the data, while a map fitted on the calibration set is not permutation-invariant in the sense required by Lemma B.2, so the intended condition should be spelled out.
[Section 3.4] The text 'Learning the entropic map estimator as in Section 3.4' is a self-reference; it should refer to Section 2.4.
[Section 4.3] The low-dimensional claim 'Results in Figure 4' should reference Figure 1; the current figure cross-references between Figures 1, 2, 4, 6, and 8 are confusing.
[Table 1] On several high-dimensional datasets the reported region sizes are astronomically large (e.g., 2E+13), which makes the region-size comparison across methods hard to interpret; a brief comment on this regime is needed.
[Throughout] There are minor typos: 'discritbution' (Section 3.1), 'encouter' (Section 3.3), 'untractable' (Remark 3.3), and 'is is' (Proposition B.1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coverage guarantee follows from conformal exchangeability and empirical quantiles, not from fitted transport-map parameters.

full rationale

The central coverage claim (Proposition 3.6, Eq. 19) is obtained by applying univariate conformal calibration to the scalarized scores ||T-hat(S(X_i,Y_i))||. The radius r-hat_{α,n+1} is defined as the empirical quantile of exactly those norms, and Lemma B.2 proves coverage by averaging indicators over the exchangeable sequence Z_1,...,Z_{n+1}. This is the standard conformal construction, so the target coverage is not an input to the definition of the map or the radius. The entropic map (Eq. 11) is an estimator imported from the external OT literature (Pooladian & Niles-Weed 2021; Cuturi 2013), and the hyperparameters ε and m affect region size but not the proof of coverage. Self-citations (Cuturi 2013; Cuturi et al. 2022; Peyré & Cuturi 2019; Guha et al. 2024) are implementation or related-work references and are not load-bearing for Proposition 3.6. The main caveat is a protocol/validity concern, not circularity: Remark 3.3 explicitly warns that 'the coverage guarantee does not hold if the transport map is computed solely on a hold-out independent dataset,' and the implementation in Section 3.4 fits the entropic map on the n calibration residuals and calibrates on the same norms, while Lemma B.2 requires either a map fixed before calibration or one that is a permutation-invariant function of the full n+1 sequence. That gap concerns whether the finite-sample guarantee transfers to the reported protocol; it does not reduce the claimed result to its own inputs, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities; it combines existing concepts (Brenier maps, entropic maps, conformal prediction) in a new way. The free parameters are the hyperparameters of the OT map estimator and the target grid structure, which are selected by hand or tuned on the benchmark.

free parameters (3)

Entropic regularization ε = 0.1
Selected from {0.001, 0.01, 0.1, 1.0} as best overall on the benchmark datasets (Section 4.2, Figure 2); affects how closely the entropic map approximates the Brenier map and the statistical/computational trade-off.
Number of target points m (sphere discretization) = 32768 = 2^15
Selected as the largest grid size tested in Figure 2; larger m improves region size but increases Sinkhorn cost (Section 3.4, Figure 2).
Target grid shell structure (nR radii, nS directions, no origin copies) = implied by n+1 = nR nS + no
Design choice for the discrete spherical target used in Proposition 3.2 and in the entropic map target sampling; not tuned in experiments but a free structural choice.

assumptions (4)

domain assumption Exchangeability of the data points (Z_1,...,Z_{n+1})
Standard conformal prediction assumption invoked in Lemma 2.1, Proposition 2.2, and Lemma B.2 to obtain finite-sample coverage.
standard math Existence of the Brenier map T* pushing the source measure to the target measure (source has a density)
Invoked in Definition 2.3 to define center-outward ranks via the optimal transport map; requires the source distribution to have a density.
ad hoc to paper Permutation invariance or symmetry of the approximation map T̂ for the transductive guarantee
Lemma B.2 requires that the empirical quantile region be symmetric in the exchangeable variables; for the practical split algorithm, T̂ must be fixed before calibration, which the paper does not explicitly separate.
ad hoc to paper The discrete spherical target with equal mass per point yields the uniform rank distribution in Equation (14)
Prop 3.2's radius formula relies on the target grid construction n+1 = nR nS + no with equal mass on each point, giving P(||U|| ≤ r_j) = (n_o + j n_S)/(n+1).

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Cite this review

Pith. "Pith review of Multivariate Conformal Prediction using Optimal Transport." pith.science (2026). https://pith.science/paper/6PQCYIWI

@misc{pith2026250203609,
  author       = {Pith},
  title        = {Pith review of: Multivariate Conformal Prediction using Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PQCYIWI}},
  note         = {Machine review of arXiv:2502.03609}
}

read the original abstract

Conformal prediction (CP) quantifies the uncertainty of machine learning models by constructing sets of plausible outputs. These sets are constructed by leveraging a so-called conformity score, a quantity computed using the input point of interest, a prediction model, and past observations. CP sets are then obtained by evaluating the conformity score of all possible outputs, and selecting them according to the rank of their scores. Due to this ranking step, most CP approaches rely on a score functions that are univariate. The challenge in extending these scores to multivariate spaces lies in the fact that no canonical order for vectors exists. To address this, we leverage a natural extension of multivariate score ranking based on optimal transport (OT). Our method, OTCP, offers a principled framework for constructing conformal prediction sets in multidimensional settings, preserving distribution-free coverage guarantees with finite data samples. We demonstrate tangible gains in a benchmark dataset of multivariate regression problems and address computational \& statistical trade-offs that arise when estimating conformity scores through OT maps.

Figures

Figures reproduced from arXiv: 2502.03609 by the authors.

**Figure 1.** We report the mean and standard error of the region size across 10 different seeds. For M-CP, we use 300 samples to compute the conditional mean, and for OT-CP, we use ε = 0.1 and 2 15 = 32768 points in the uniform target measure. Overall, OT-CP displays smaller region size than other baselines (13 out of 17 datasets). The output dimension d of each dataset is provided next to its name. the review provided in (Nguye… view at source ↗

**Figure 2.** This plot details the impact of the two important hyperparameters one needs to set in OT-CP: number of target points m sampled from the uniform ball and the ε regularization level. As can be seen, larger sample size m improves region size (smaller the better) for roughly all datasets and regularization strengths. On the other hand, one must tune ε to operate at a suitable regime: not too low, which results in the we… view at source ↗

**Figure 4.** As in 1, we report mean and standard errors for region size (log scale) across 10 different seeds for larger datasets. We keep the same parameters and importantly ε = 0.1 and 2 15 = 32768 points in the uniform target measure. We expect the performance of OT-CP to decrease with dimensionality, but it does provide a convincing alternative to the other approaches. lenges brought forward by the estimation of OT maps in… view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: Conformal sets recovered by mapping back the reduced sphere on the Manhattan map, in agreement with Equation 18, on a prediction for the taxi dataset. We use the inverse entropic map mentioned in Section 3.4, mapping back the gridded sphere of size m = 215 for each lev…

**Figure 6.** Figure 6: Coverage for higher dimensional datasets, corresponding to the setting displayed in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Runtimes for higher dimensional datasets, corresponding to the setting displayed in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Ablation: coverage quality as a function of hyperparameters, with the setting corresponding to [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Coverage of all baselines on small dimensional datasets, corresponding to the region sizes given in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: Ablation: running time as a function of hyperparameters, with the setting corresponding to [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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