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Optimal Transport-based Conformal Prediction

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper introduces OT-CP, a conformal prediction method that builds prediction regions for multivariate nonconformity scores via Monge-Kantorovich ranks and quantiles, and proves finite-sample distribution-free coverage under…

desk verdict OT-CP is a genuinely useful combination and the lower-bound coverage proof is sound; the printed two-sided bound has a real but fixable formal defect. read the letter →

arxiv 2501.18991 v2 pith:PNGZJHKK submitted 2025-01-31 stat.ML cs.LG

Gauthier Thurin (CNRS) , Kimia Nadjahi (CNRS) , Claire Boyer (LMO) This is my paper · ORCID
classification stat.MLcs.LG MSC 62G1562G20
keywords optimaltransportconformalpredictionMonge-Kantorovichranksmultivariatequantilesdistribution-freecoveragemulti-outputregressionmulticlassclassificationconditional
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

The paper introduces OT-CP, a conformal prediction method that handles multivariate nonconformity scores by ranking them with optimal transport. It proves that the resulting prediction regions achieve finite-sample, distribution-free coverage under exchangeability, with the coverage gap controlled by the maximum number of ties in the transport ranks. Because the regions are sublevel sets of multivariate ranks rather than predefined balls, boxes, or ellipsoids, they can follow the actual geometry of the data errors. The paper works out the procedure for multi-output regression and multiclass classification, and adds an adaptive variant with asymptotic conditional coverage.

What carries the argument

The empirical Monge-Kantorovich rank map $R_{n_1}(s) = \operatorname{argmax}_{U_i}\{\langle U_i, s\rangle - \psi_{n_1}(U_i)\}$, where the $U_i$ are reference rank vectors on the sphere and $\psi_{n_1}$ solves the dual Kantorovich problem, assigns each score a transport-based rank. Ordering scores by $\lVert R_{n_1}(s)\rVert$ turns a level set of these rank norms into the quantile region $\hat{Q}_n(\alpha)$. The load-bearing device is the split: the rank map is built on $D_1$, the threshold quantile is taken on $D_2$, so the rank of the test score is exchangeable with the $D_2$ ranks and the standard quantile lemma applies. A second device, choosing the reference distribution on the positive orthant, realigns the ordering for absolute-error classification scores.

What would settle it

One concrete check: take a discrete score distribution with atoms on the boundary cells of the empirical transport map, choose any deterministic tie-breaking rule, and test whether empirical coverage over many splits stays within $\alpha$ and $\alpha + n_{\text{ties}}/(n_2+1)$; a rule that makes coverage fall below $\alpha$ on such data would refute the 'any distribution' reading of the theorem.

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Extended reading notes

Core claim

The paper's central claim is that conformal prediction can be extended to multivariate nonconformity scores without losing distribution-free finite-sample coverage, by replacing scalar quantiles with Monge-Kantorovich (MK) ranks and quantiles. The prediction region is the set of scores whose MK rank norm falls below the $\lceil (n_2+1)\alpha\rceil$-th order statistic of calibration ranks computed on a separate split, and Theorem 2.4 bounds its coverage between $\alpha$ and $\alpha + n_{\text{ties}}/(n_2+1)$ for exchangeable data. Because the MK rank map is built from an optimal transport coupling between the score distribution and a reference distribution on the sphere, the resulting regions inherit the shape of the empirical score distribution and can be non-convex. The paper also shows that choosing a reference distribution on the positive orthant gives a left-to-right order suited to classification scores, and that a k-nearest-neighbor conditional version of the rank map yields asymptotically valid conditional coverage.

Load-bearing premise

The rank map's argmax defines a single rank for every score, but for scores lying on cell boundaries in the transport geometry, or for discrete score distributions, this argmax can be multi-valued; the paper specifies no tie-breaking rule, yet the coverage guarantee and the prediction region require one.

Editorial extensions

If this is right

  • Prediction regions for multi-output regression can be non-convex and mirror the residual geometry while still satisfying the finite-sample coverage bound.
  • For multiclass classification, using the full softmax score vector gives label-conditional coverage comparable to adaptive scores while retaining the efficiency of simpler scores.
  • The adaptive OT-CP+ variant with k-nearest-neighbor conditional rank maps achieves asymptotic conditional coverage under mild density assumptions.
  • When the score is univariate, the optimal transport problem reduces to sorting and OT-CP recovers standard conformal prediction at $O(n \log n)$ cost.
  • With a specified tie-breaking rule, the upper coverage gap becomes $1/(n_2+1)$, matching the usual conformal guarantee up to discretization.

Reading between the lines

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

  • The coverage theorem is silent about how to resolve ties when the argmax in the rank map is set-valued, so the 'any distribution' claim implicitly depends on a tie-breaking convention; a careful empirical study on discrete or boundary-atom score distributions would show whether coverage stays within the stated bounds for each convention.
  • The splitting strategy makes the transport rank of a test point distribution-free, so it could be reused for other transport-based ranks or quantile maps beyond the Monge-Kantorovich construction.
  • A natural extension is to align the reference distribution with the geometry of the task, for instance using the positive-orthant reference for multi-label or multi-hot scores, which the paper suggests as a direction.
  • Replacing the exact OT solver with entropic approximations could cut the $O(n^3)$ cost and make OT-CP practical for large calibration sets, at the price of a bias that the current theory does not cover.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes OT-CP, a split-conformal prediction method that handles multivariate non-conformity scores by using empirical Monge-Kantorovich (MK) vector ranks and quantiles built on optimal transport. The procedure constructs flexible, potentially non-convex prediction regions, and the authors claim a finite-sample distribution-free coverage guarantee in Theorem 2.4. They also introduce an adaptive variant, OT-CP+, for multi-output regression with an asymptotic conditional coverage guarantee, and an adaptation to multiclass classification. The numerical experiments compare the proposed methods against ellipsoidal, rectangular, and scalar-score baselines on simulated and real data.

Significance. If the coverage guarantee is stated and proved correctly, the paper makes a valuable contribution: it provides a principled way to use multivariate scores in conformal prediction without imposing convex or predefined set shapes, while retaining a finite-sample, distribution-free validity statement. The explicit split of the calibration set to handle the dependence introduced by the estimated transport rank map is a sensible and nontrivial device. The OT-CP+ extension toward conditional coverage and the classification experiments with worst-slice and label-wise coverage metrics give the paper useful practical scope. The theoretical results are not machine-checked and no code is released, but the proofs are sufficiently detailed to allow verification of the main mechanism.

major comments (2)
  1. [Theorem 2.4, Eq. (6), and Appendix B.1] The two-sided bound is not a well-formed scalar probability inequality as stated. The quantity nties is defined as the maximum number of ties in the realized sample {∥Rn1(s(X_i,Y_i))∥ : (X_i,Y_i) ∈ D2} ∪ {∥Rn1(Stest)∥}, so it is a random variable depending on the data, whereas P(Ytest ∈ Cα(Xtest)) is a fixed real number. In the proof of Lemma B.1, the step bounding the count of entries ≤ U(k) by k−1+nties is only valid for a realized tie count; after taking expectations the upper bound must involve E[nties] (or an almost-sure upper bound), not nties itself. The same issue transfers to Eq. (6). The lower bound α is unaffected, but the claimed sharp upper bound needs to be restated either conditionally on the realized tie pattern or as a marginal bound with E[nties]. This is a load-bearing correction to the central theorem.
  2. [Eq. (3) and Section 2.3, step 2] The empirical MK rank map is defined as an argmax over the finite reference set {U_i}. For scores lying on cell boundaries of the piecewise-linear convex potential, or for discrete score distributions, this argmax can be set-valued, and the paper specifies no tie-breaking rule. Since the prediction region in Eq. (5) and the coverage event in Theorem 2.4 are only defined for a single-valued rank map, the claim that the guarantee holds 'for any score' silently presupposes a deterministic selection rule. A concrete rule (e.g., lexicographic or smallest-index selection) should be stated, and the proof adjusted accordingly; the lower-bound argument is robust to this choice, but the theorem is not fully well-defined without it.
minor comments (4)
  1. [Section 2.4, paragraph after Theorem 2.4] The sentence 'a tie-breaking rule can be applied if ties occur, as usually done in CP, to enforce nties = 1' is imprecise: tie-breaking in the ordering of equal-norm ranks does not change the multiplicity of identical values in the realized sample, so it cannot by itself force nties = 1 for all score distributions.
  2. [Definition 2.1 and Section 2.3, step 2] The notation Rn1 used in step 2 is not formally introduced; Definition 2.1 defines Rn for a sample of size n. Please state explicitly that Rn1 denotes the empirical MK rank map computed from D1, with reference vectors {U_i}_{i=1}^{n1}.
  3. [Section 4, Figure 5] The captions of Figure 5 and the surrounding text use 'center-outward' versus 'left-to-right' ordering before the positive reference simplex is defined; a one-sentence precise definition of the positive reference rank vectors in the main text would improve readability.
  4. [Appendix B.3, proof of Lemma B.2] The uniform convergence of the empirical quantile function is invoked via Bogoya et al. (2016); please state the exact conditions (boundedness and continuity of the limiting quantile) that justify this step, since the argument otherwise relies on an implicit regularity assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coverage guarantee follows from exchangeability and the standard quantile lemma, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim, Theorem 2.4, is a split-conformal coverage bound. The MK rank map is computed on D1, then the scalar values ∥Rn1(·)∥ for D2 and the test point are exchangeable by a standard external result (Kuchibhotla 2020, Proposition 3), and the quantile lemma (Lemma B.1) yields the probability bounds. The MK rank construction ensures that the calibration ranks are a permutation of the reference vectors, which is a mathematical property rather than a fitted parameter; the threshold ρ is an order statistic of D2, not optimized to match the test outcome. The asymptotic conditional coverage result (Theorem 3.2) imports consistency of conditional MK quantile regression from del Barrio et al. (2024), an external source with no author overlap with the present paper. No parameter is fitted to the calibration data and then renamed as a prediction, and no load-bearing self-citation appears. The data-dependence of nties in the upper bound of Theorem 2.4 is a formal correctness concern about the statement of the bound, not a circularity: the derivation does not assume the conclusion it aims to prove. The derivation is therefore self-contained with respect to its stated inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central finite-sample coverage result relies only on exchangeability and a fixed score function; it does not require fitting any parameter. The conditional extension OT-CP+ introduces a kNN hyperparameter and inherits consistency assumptions from del Barrio et al. (2024). No new physical or model entities are postulated.

free parameters (1)
  • k (number of neighbors in OT-CP+) = n/10 or sqrt(n) depending on dataset (Appendix D.2)
    Hand-chosen hyperparameter in the conditional variant. The asymptotic theorem requires any sequence with k → ∞ and k/n1 → 0, so the guarantee does not hinge on the exact value.
assumptions (5)
  • domain assumption Exchangeability of calibration and test data
    Theorem 2.4 explicitly assumes {(X_i,Y_i)} ∪ (X_test,Y_test) exchangeable (Section 2.4).
  • domain assumption Score function is fixed and independent of calibration data
    OT-CP assumes a given score s(x,y); if s were selected using D2, the exchangeability of the rank norms would fail. This is implicit in Section 2.1.
  • ad hoc to paper The MK rank map argmax in Eq. (3) has a well-defined single-valued selection
    For discrete score distributions or boundary scores, the argmax over the finite reference set can be non-unique; no tie-breaking rule is specified, yet the theorem claims validity for any score distribution.
  • domain assumption Consistency of conditional MK quantile regression (del Barrio et al., 2024, Corollary 3.4)
    Theorem 3.2's asymptotic conditional coverage rests on this external result, together with Assumption 3.1 on the conditional density.
  • standard math Exchangeability of ranks under a fixed rank map (Kuchibhotla, 2020, Proposition 3)
    Used in the proof of Theorem 2.4 to justify that rank norms of D2 and the test point are exchangeable given D1.

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Cite this review

Pith. "Pith review of Optimal Transport-based Conformal Prediction." pith.science (2026). https://pith.science/paper/PNGZJHKK

@misc{pith2026250118991,
  author       = {Pith},
  title        = {Pith review of: Optimal Transport-based Conformal Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNGZJHKK}},
  note         = {Machine review of arXiv:2501.18991}
}
read the original abstract

Conformal Prediction (CP) is a principled framework for quantifying uncertainty in blackbox learning models, by constructing prediction sets with finite-sample coverage guarantees. Traditional approaches rely on scalar nonconformity scores, which fail to fully exploit the geometric structure of multivariate outputs, such as in multi-output regression or multiclass classification. Recent methods addressing this limitation impose predefined convex shapes for the prediction sets, potentially misaligning with the intrinsic data geometry. We introduce a novel CP procedure handling multivariate score functions through the lens of optimal transport. Specifically, we leverage Monge-Kantorovich vector ranks and quantiles to construct prediction region with flexible, potentially non-convex shapes, better suited to the complex uncertainty patterns encountered in multivariate learning tasks. We prove that our approach ensures finite-sample, distribution-free coverage properties, similar to typical CP methods. We then adapt our method for multi-output regression and multiclass classification, and also propose simple adjustments to generate adaptive prediction regions with asymptotic conditional coverage guarantees. Finally, we evaluate our method on practical regression and classification problems, illustrating its advantages in terms of (conditional) coverage and efficiency.

Figures

Figures reproduced from arXiv: 2501.18991 by the authors.

Figure 1
Figure 1. Ranking multivariate scores using optimal transport. The colormap encodes how the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Conformal multi-output regression with OT-CP on simulated data [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Adaptive conformal regression with OT-CP+ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Conditional coverage on real datasets of two adaptive conformal procedures for multi [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Ordering must depend on the chosen scores: (a) Center-outward for signed errors, (b) [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Positive reference ranks for a left-to-right ordering. The colormap encodes how the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Conformal classification by Quadratic Discriminant Analysis on simulated data [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Classification metrics on MNIST and Fashion-MNIST, results averaged over the 10 labels [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Label-wise coverage on K = 10 classes of MNIST and Fashion-MNIST which already shows several benefits while remaining conceptually simple. 5 Conclusion and perspectives We have introduced a general and versatile framework for conformal prediction grounded in optimal tr…
Figure 10
Figure 10. Figure 10: Label-wise results on K = 10 classes of MNIST D.2 Implementation details for regression In [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Label-wise results on K = 10 classes of Fashion-MNIST 1000 2000 3000 4000 5000 6000 Size of calibration data 0 20 40 60 80 100 Time (seconds) OT-CP OT-CP+ [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: Computational time for OT-CP and OT-CP+ against the number of calibration instances [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Marginal coverage, volume and calibration time (in seconds) for experiments of Figure [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: Classification by Quadratic Discriminant Analysis on simulated data, with a different [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: Label-wise results on K = 5 classes of MNIST 0 1 2 3 4 Label 0.5 0.6 0.7 0.8 0.9 1.0 Coverage Method IP MS APS OTCP (a) Coverage 0 1 2 3 4 Label 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Size Method IP MS APS OTCP (b) Average size 0 1 2 3 4 Label 0.0 0.2 0.4 0.6 0.8 1.0…
Figure 16
Figure 16. Figure 16: Label-wise results on K = 5 classes of Fashion-MNIST 27 [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]

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