Neural Optimal Transport Meets Multivariate Conformal Prediction
Authors: Vladimir Kondratyev
Alexander Fishkov
Mahmoud Hegazy
Nikita Kotelevskii
Rémi Flamary
Maxim Panov
Eric Moulines
Venue: ICLR 2026 Poster (2026)
Keywords: vector quantile regression
conformal prediction
neural optimal transport
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: We propose a framework for conditional vector quantile regression (CVQR) that combines neural optimal transport with amortized optimization, and apply it to multivariate conformal prediction. Classical quantile regression does not extend naturally to multivariate responses, while existing approaches often ignore the geometry of joint distributions. Our method parameterizes the conditional vector quantile function as the gradient of a convex potential implemented by an input-convex neural network, ensuring monotonicity and uniform ranks. To reduce the cost of solving high-dimensional variational problems, we introduce amortized optimization of the dual potentials, yielding efficient training and faster inference.  

  We then exploit the induced multivariate ranks for conformal prediction, constructing distribution-free predictive regions with finite-sample validity. Unlike coordinatewise methods, our approach adapts to the geometry of the conditional distribution, producing tighter and more informative regions. Experiments on benchmark datasets show improved coverage–efficiency trade-offs compared to baselines, highlighting the benefits of integrating neural optimal transport with conformal prediction.

NEURAL OPTIMAL TRANSPORT MEETS MULTIVARIATE CONFORMAL PREDICTION
Vladimir Kondratyev
Alexander Fishkov
Department of MI., MBZUAL, UAE v1 vladimir .e
Department of MI., MBZUAT, UAE
Mahmoud Hegazy
Nikita Kotelevskii
CMAP, École polytechnique, France
Department of ML, MBZUAL, UAE
Rémi Flamary
Maxim Panov
CMAP, Ecole Polytechnique, Palaiseau, France
Department of MI, MBZUAL, UAE
Eric Moulines
CMAP, Ecole Polytechnique, Palaiseau, France Les zuai ac.ac ae
ABSTRACT
Classical quantile regression does not extend naturally to multivariate responses, while existing approaches often ignore the geometry of joint distributions. We propose a framework for vector quantile regression (VQR) that combines neural optimal transport with amortized optimization, and apply it to multivariate con- formal prediction. We then exploit the induced multivariate ranks for conformal prediction, constructing distribution-fres predictive regions with Enite-sample lidity. Unlike coordinatewise methods, our approach adapts to the geometry of the conditional distribution, producing tighter and more informative regions. Experi- ments on benchmark datasets show improved evverae--ffficenne trade-offs com- pared basclines, highlighting the benefits of integrating neural optimal transport with conformal prediction.
INTRODUCTION
Quantile regression has long been comnerstone for modeling heterogeneous conditional distribu- tions in the univariate setting (Koenker & Bassett 1978 Koenker] 2005). By estimating conditional quantiles rather than conditional means, it captures heteroscedasticity, asymmetry, and tail behav- ior, and the collection of conditional quantiles provides a detailed description of the conditional law of response variable. This perspective has supported broad range of advances in econometrics, statistics, and machine learning.
Extending these ideas multivariate responses, however, remains challenging. Unlike the scalar case, Rd lacks a natural total ordering. As result, early multivariate notions of quantiles-basse based on projections, spatial medians, or depth functions prociidns-provide useful summaries but generally inherit only part of the desirable scalar properties (Chaudhuri) 1996 Hallin et al. 2021).
Recent progress in optimal transport has offered a principled definition of multivariate ranks and quantiles (Chernozhukov et al.) 2017: Hallin & Konen 2024 Peyré al. (019).. By interpreting quantiles as transport maps from a reference distribution to the law Y. these approaches recover distribution-free center-outward ranks and nested quantile regions that extend univariate order statis- ties to high dimensions. Building on this perspective, vector quantile regression (VQR; Carlier et al.) 2016 2017) introduces conditional vector quantile functions (CVQFs), cyclically monotone maps that represent Y as transformation of latent uniform variables given covariates. CVQFs provide a rich yet tractable representation of conditional distributions, and recent work has developed practical

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estimators and extensions beyond linear models (Rosenberg et al. 2023 Vedula et al. 2023b Sun
iet al. 2022 del Barrio et al. 2025).
In parallel, conformal prediction provides general framework for constructing predictive regions with samile marginal coverage under exchangeability (Shater & Vovk 2008 Angelopoulos et al. 2023). Although well studied in the univariate case, multivariate extensions
less devel- oped, and many existing constructions do not explicitly exploit the geometry of the joint conditional distribution. Existing approaches include coordinate-wise methods that treat each coordinate sepa- rately (Zhou et al. 2024, Diquigiovanni et al., (20T); I; reductions of the multivariate problem to one dimension via a scalar score (Dheur al.J 2025; Izbicki al. 2019); and structured approaches that use deep generative embeddings (Dheur et al., 2025 Feldman et al.) [2023). Many of these methods rely on heuristic choices of scores or shapes, with limited theoretical guidance on how to encode multivariate dependence. For an overview of recent developments in multivariate confor- mal regression, see (Dheur et al. (025). Very recent work begins to bridge this gap by incorpo- rating optimal-transport based multivariate ranks into conformal prediction, yielding theoretically grounded multivariate prediction sets (Thurin et al. 2025 Klein et al. 2025).
In this paper, we leverage neural optimal transport framework for learning CVQFs (Seguy et al., 2018 Makkuva et al. 2020), which allows us to estimate parametric cyclically monotone vector quantiles and multivariate ranks. Building the resulting multivariate ranks, use conformal prediction to produce distribution-fre valid prediction regions that adapt to the geometry of condi- tional distributions in the multivariate setting.
We make three main contributions:
We develop a neural optimal transport framework for learning vector quantile regres- sion (VQR) and the associated multivariate ranks (Section 3)
We provide a principled integration of vector quantiles and multivariate ranks into conformal prediction, yielding distribution-free predictive regions with smitee marginal coverage that adapt to the geometry of conditional distributions (Section 4
We show in numerical experiments that the resulting conformal prediction sets consis- tently improve calibration and reduce set size relative to coordinatewise and representation- based baselines across benchmarks (Section 6).
CONSTRUCTING MULTIVARIATE CONFIDENCE SETS
We begin by informally introducing the conditional vector quantile and rank maps, which provide a flexible representation of the conditional law of Y given X.
Quantiles in ID and Confidence Sets. Let us first consider the case of Y E E C R. Let XY ~ Fyx and let Fyjx be the conditional distribution of Y given X. For any a E [0. and any x E X, the quantile function (, returns the quantile value Qri (a,, x) E y of the conditional distribution Fyxx=1. Knowing the quantile function is instrumental for constructing confidence sets. For example, for given a (0,1) one define Ca (x) (Qrix(aa2,)) Qr a)2. Under standard regularity conditions (e.g., continuity of Fyjx=s), this confidence set is conditionally valid, i.e. P(Y E C. (x) X x) = -1-60 a.
The inverse map Q is sometimes called a rank function: for any value it produces a value (v, E [0, 1). which can be interpreted as the rank of y under the distribution Fy|x=== Importantly, the probability integral transform implies that the distribution of (Y, X is uniform on [0. 1]. In turn, the rank function gives an equivalent way to define the confidence interval, C. (x) = {y: Qyix (v) E [a/2, 1 - a/2]]. This shows two interchangeable viewpoints: one may construct C. (ar) either from the quantile function from the rank function.
Multivariate Quantiles. In the absence of a natural order on Rd for > 1, the definition of multivariate quantiles is nontrivial. In this paper, we study multivariate quantiles based on optimal transport: see among others (Carlier et al. 2016 Hallin et al., [02TT] Hallin & Konen, (0224) Peyré et al. 2019). We start by looking at specific example, while the full exposition is given Section
2

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Let Volume denote the Lebesgue volume normalized on 2 == B(0, 1) that Volume(44) 1. Define T1-a E Rt such that the Euclidean ball r1-aa C U satisfies the condition Volume(B(0,7)
= a. Then, it can be shown (see Theorem [ below) that there ex- ists a map (u,x(u, and a uniform random variable U over U, independent of X, such that Y = Qrix (U) almost surely. This map is called a vector quantile. The corresponding inverse map Qrx (v. x) U is a natural analogue of the rank function and yields multivariate ranks that uniform on U conditional on X.
We can directly proceed with the construction of confidence sets based on Qyix For a E X, define the pullback set
CD (x) = {y: Qyix (v, E B(0,)=))))
Using the key identity (9yx(
U almost surely, we get that
P(Y E (X))
= =FW. (11 XII sm = =FU~Fo (1U S r-
(2)
Hence, by the definition of ri the coverage of the pullback set CPb (x) is exactly 1- a. Moreover, o
since U is independent of X, the same argument yields P(Y Cpb (x) X x) = 1 o for each x X.
Conformalized Confidence Sets. In practice, we can only have access to the estimator Qylx of Qvix One can consider plug-in confidence sets constructed by replacing Qvix with Qyix in However, such sets do not, in general, guarantee coverage, because typically Qyix # Qvix Con- sequently, the coverage of Cpb(X) may be miscalibrated, motivating the use of conformal predic- tion (Shafer & Vovk 2008 [omano al.) [209)] /Angelopoulos et al., [223). Conformal prediction corrects this miscalibration and provides finite-sample, distribution marginal coverage guaran-
tees under exchangeability.
Specifically, given calibration set Dent = {(,, YD independent of the training data, consider score S; 1yr) (Y, i l,...; : n. Let Úi-o be the empirical (1 -1) + -uuanille of {S:]i= Then split-conformal prediction constructs, for a new covariate value X the set
(Xrett) == {y E 11 (v, Xteat) S 91-a]
such that P(Testt (X1, Yi).... (Xn, Yn), (Xtest, Yrest.) are exchangeable (Romano et al. 2019 Angelopoulos al., 2023). Other choices of the score are possible, see discussion in Section what follows,
Čeb(Xtest))
a, under the assumption that
we discuss various approaches to construct Qyix and Qy based on neural optimal transport.
VECTOR QUANTILE REGRESSION VIA OPTIMAL TRANSPORT
We recall the optimal-transpon construction of conditional vector quantiles and multidimensional ranks. Our presentation follows closely the formulation of Carlier et al. (2016); Hallin et al. (202I); see also Brenier (199I); Santambrogio 2015)) for background on Brenier maps and optimal trans- port, and Chermnzzuuoo al. (2017) for related OT-based multivariate ranks and quantiles.
Let (Y, (Y,X) a random vector on a complete probability space (St, A, IP,, where Y E Rd, and X E Rd. Throughout this section, set Fyix the conditional law of Y given X, and by Fx the marginal of X. Let U be a random vector on (S,,A,)) IP) with reference distribution Fr. We write V,K,U, X,U X for the supports of Fy,Fyx Fx, Fr. Fyx, Fux, and Vd for the support of Fyix=x- (Here, y x X and U x X are not assumed to coincide with Cartesian products of marginal supports; they are simply notational shorthands for the supports of the corresponding joint laws.) By U < for U. Rd mean coordinatewise inequality between the vectors, that is, U, S uy for all coordinates j E {1.....d) d). This induces the multivariate CDF notation (u) Fe Fu(w1,.- (#1,...: ua) P(U, S uu...., Ua S
dy. Denote by Fyx the joint law of (Y, by
3

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ud) PU I u). All norms || || are Euclidean R The following properties of Fu and Fylx are sufficient for the construction of OT-based conditional vector quantiles and ranks. Assumption 1. The reference distribution Fy admits a density fo with respect to Lebesgue measure on R,, with convex support U CRa
Typical choices for Py include the uniform distribution on [0, 1j9. the Gaussian (N Ia), or any distribution on Re with convex support.
Assumption 2. For each a E X. the conditional law ()) has density Jy fy x))..
Our aim is construct a measurable map that transports Fū the conditional law of Y given X, with multivariate monotonicity property. In dimension d >> 1, the appropriate notion of monotonic- ity is given by gradients of convex functions (equivalently, eyclical monotonicity for the quadratic cost), which motivates the conditional vector quantile function (CVQF).
Theorem (Carlier et al. 2016), Theorems 2.1 & 2.2). Suppose Assumption holds. Then:
(i) For each X, there exists measurable map the gradient of a convex function and pushes FU forward to FyJx=r
Qrix (,, unique Fv-a.e., which is
(i) Consequently, there exists U ~ Fy such that Y = Qvix X) a.s. with U | X ~ Fu.
(ii) Additionally, if Asssupption 2 holds, then there exists measurable inverse map Qyix (v, u such that Qrix (Qy]x (U,)x((,x))) (u, x). for Fu-a.e u, and P(Qy)xpQXXXX (X, S X = =x)
(u).
The map y Qyix(1 (3) is the conditional vector rank. When distribution on [0, 1], this rank coincides (a.e.) with the conditional CDF. For d > 1, there is no analogous identification with coordinatewise conditional CDF, and the rank is intrinsically vector- valued (Hallin et al. 2021 Hallin & Konen 2024 del Barrio et al. (025).
and Fy is the uniform
Finally, the following moment condition ensures that the quadratic OT objective is well-defined and that the duality-based characterization below applies.
Assumption 3. Y and U have finite second momenis: [[[[|||] < oo and [[[ < co.
Under Assumptions and the CVQF admits the following optimal-transport transport characterization:
min E E| - V]["] s.t. V X~ Fu,
The dual program is
min E (V, X)] + [[ X)] vi.
s.t. p(u, x) + (v, 2 uTT.
(3)
where V is any vector such that V X Fr. and the inequality is required for all E X, u, and y E Vd. The following properties of the solution to (3) hold.
Theorem (Carlier et al. (2016), Theorem 2.3). Suppose Assumptions
73 hold. Then,
(i) There exist potentials x((,x) and x)) sp (y, solving 3 where for each x, u +
( x)) and y x)) are convex and Legendre conjugates:
p(u,x) sup(u y x)),
yEy
(4)
v(y,x) 40 (y. x)) sup v(u,x))) well
(i) The conditional vector quantile is Qv)x((,xx) = Vuplu, x)) for Fu-ae. u.
(ii)) The conditional vector rank is Q x) V I) for Fy y.
(iv) These maps are inverses: for each x, V Vy (Vap(u x), x) = u. Vup(( (oo x), = y. for
Fu-a.e u and F (,, ()..e...e y.
This characterization reduces the construction of Qy and Qrlx to finding convex dual potentials solving (3). and it underlies computational approaches based on convex optimization and discretized optimal transport.
4

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We now describe practical approach for learning continuous Neural VQR models. We first rewrite the optimization problem using a single convex potential via the conditional c-transform. We then discuss implementable parameterizations with Partially Input Convex Neural Networks (PICNNs; Amos et al., 2017] [Bunne et al. 2022) and two standard strategies to reduce training cost: amortized optimization and entropic variant.
Neural parameterization and semi-dual formulation. First, following Taghvaci & Jalali (2019)
Makkuva al. (2020); Amos (2023), we reformulate the Konge-Kantoroviie dual problem as an optimization over a parametric family of potentials yoo with parameters 0. We enforce that 50 is
convex its first argument, so that the Fenchel-Legende conjugacy in (the c-transform) yields
a unique conjugate potential.
For each E X introduce the conjugate of pointwise potential (p((;.)): + y
(0((22) v) u po(u, x); x)) Jep((--) (uppll (1), (pp( (v) = arg max J pp(( mn) (u. y). wEll
(5)
(6)
(7)
With these notations, the problem 3 be reformulated as the minimization of V(0), defined as
((0) == E( [U)( + +Byxx))-FY [x)
(8)
Here, Fy Fx denotes the product measure of Fo and Fx, corresponding to independent sampling of U ~ Fu and X Fx. The optimal parameter can then be found by gradient steps on ((0). By
Danskin's theorem (Danskin 1967). differentiating through the conjugate only requires derivatives of ypa and yields
(OV B(U.))FFFFFX (o,o(U,))) [(rx))=Fyx Vopolu, X) (o
Remark 1. Above we focus on the dual potential a=) which is linked to Fr, while its conjugate
()) is linked to Fyix( (- X = x). Due to the symmetry of (3), one may instead parameterize #)(;; ; our experiments we consider both choices.
Neural Quantile Regression with PICNNs (C-NQR). Convexity of (o((,.)) with respect to the first argument can be achieved (Bunne et al. 2022) using PICNNs (Amos et al. (017). The main computational challenge in 8 is that the conjugate (),(.)) must be computed for each æ in the mini-batch. As first strategy, we compute the argmax in (7) exactly with an L-BFGS solver (Liu & Nocedal 1989). and update the PICNN parameters with stochastic gradient descent; see Algo- rithm I Appendix [E66] We refer to this method conditional neural quantile regression (C-NQR). We distinguish C-NQR, and C-NQRy depending on whether we estimate şo or, equivalently by symmetry, the. This approach is straightforward but be expensive because it requires repeated inner optimization, especially for large mini-batches or high- digh-dimensional data.
Amortized Neural Quantile Regression (AC-NQR). To reduce the of repeatedly solving
we use amortized optimization, following Amos 2023). We introduce an amortization model i (y, 2) parameterized by & that predicts approximate maximizer:
(. 2)) (v)
We focus amortization architectures based on PICNNs, as in MMakkuva et al. 2020 Korotin al. 2019). The amortization model is trained jointly with pp using a quadratic loss encouraging (3 x) match (.a(t.)) (v); see Algorithm 2 in Appendix E.6 E.66 We follow a two-time-scale train- ing scheme (Konda & Tsitsiklis 2004 Borkar 2008), where the amortization model is updated on a faster timescale than sps. We refer to this method amortized conditional neural quantile regression (AC-NQR), with AC-NQRu and AC-NQRy defined analogously to C-NQR.
Entropic regularized Neural Quantile Regression (EC-NQR). The two approaches above still require solving a convex optimization problem to compute (approximate) conjugates, which can become costly in high dimensions. An alternative is to entropic regularization, which enables stochastic gradient solvers that scale better (Cuturi 2013 Genevay et al. 2016), but introduces bias
that distort the geometry of quantile maps (Rosenberg et al. 2023). Neural parameterizations
5

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of entropic OT dual potentials studied in the non-conditional case by Seguy et al. (2018)
extend this idea to the conditional Neural VQR setting.
Concretely, we add entropic regularization term to the primal OT problem. This smooths the ob- jective and yields closed-form surrogate for the conjugate (the argmax in (7) becomes a softmax). We refer to this method as entropic conditional neural quantile regression (EC-NQR). The result- ing objective replaces inner conjugate optimization by an expectation that can be approximated by sampling; see Algorithm
3 in Appendix E.6 Additional details are provided in Appendix and
Appendix [
Remark 2. By parameterizing the dual potential as convex neural network, we restrict the model class. This raises the question of whether a more flexible non-convex generative model, such as a normalizing flow that parameterizes bijection, could replace the convex potential while still defin- ing a meaningful quantile operator. We show that this not the case and provide a counterexample in Appendix G In particular, cyclic monotonicity of the transformation is essential for statistically meaningful notion of multivariate rank.
CONFORMAL, PREDICTION WITH OT NEURAL MAPS
In this section, we demonstrate the use of our neural OT framework constructing intrinsically adaptive confidence sets with CP. The key idea is exploit multivariate quantile and rank maps learned by conditional neural OT as a building block for defining conformity scores and constructing calibrated prediction regions. Let (Y, X) ~ Fyx and E (0, 1) and denote by Qyix proxy for the true associated vector rank function Qrix as in Theorem
Generalizing conformalized quantile regression. In the univariate case, conformalized quantile 2019) replaces nominal quantile with the empirical (1 a)-
regression (CQR: Romano quantile of residuals, ensuring distribution-free, finite-sample coverage. The same principle extends the plug-in pullback set in ① Define conformity scores:
S = (Y,,
(Y1, E Deal-
Let S(1) .... < S(n) denote the order statistics, set A = [n + 1)(1 a]], and P1-a = S(k)- The conformal set
Cgb(a)) {y: (v, ((0.
then guarantees P(r:)) (Y E (x)( 1 a. We now show that this construction of confidence sets is optimal when the Jacobian of the inverse transport admits radial structure.
Theorem (Volume-optimality of pullback balls under radiality). Fix X and reference distri- bution Fu (u) = @([[ul)) for a strictly decreasing o: [0, oo) + (0, co) on U, under the assumptions of Theorem let Qrix and Qrix be the vector quantile and multivariate rank functions. Assume that there exists ju such that for all y in the support of Fy|x, it holds
det V ( jz (1QVixx(.x)11)
and the function + Φ(r) (r) is strictly decreasing, Let O be the unique radius satisfying yi u(B..) = a, where y the law corresponding to Fú and B, {u: ||u]]| S r). Define the pullback ball Cpb (x) = {y: (1, S sra.. Then, Cph(x) minimizes Lebesgue volume among all sets with r-conditional coverage of at least 1 a, ie., for every measurable A C Ya with P(Y E A X = x} 2 a, a, Vol(Ceb(() < Vol(A).
Equivalently, Theorem 3 shows that CDb (x) is the highest probability density (HPD) region for Y X at level 1 a. A noteworthy specialization, where the assumptions of Theorem are met, is the elliptical case (including Gaussian) with Fyix and Fy belonging to the same elliptical family. We defer the proof and additional details to Appendix
Re-ranked pullback sets. This construction is effective only if the scores S, capture isotropic structure. Indeed, Cpb (x) is the preimage of a centered Euclidean ball in U, implicitly assuming that the conditional distribution of U Qy (Y.X) is radially symmetric. When Qy is misspecified,

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however, the ranks may be anisotropic, and Euclidean radii become unreliable. We note that the vector ranks {U; = Qy (Y, X,) can themselves be interpreted multivariate score functions and as such be combined with the OT-CP approach of (Thurin et al. (2025). which is designed to ' be reranking approach, designed to correct deviations from reference distribution Fv. Reranking induces new scores S7
conformalize multivariate score functions. In particular, let R: U
[R)) As before let ST ... S (()) denote the order statisties, set k [(n + 1) - a} and pi = S Then, the conformalization step applied to the adjusted scores yields a calibrated radius and the prediction set
(k)
Crpb (x) {v: R RQQ/Vixx--O0 (v. Ộ(1 -a)}
where ²(1 a)) = (u: || u)ll S p. See additional implementation details in Appendix E.7 Remark 3. For completeness, we also consider a complementary construction that leverages the OT quantile and rank maps to estimate the conditional density via the change of variables formula. Using the estimated density as conformal score, this approach yields valid regions and can capture disconnected geometry when Fyix=r multimodal, e.g. Gaussian mixture. We provide additional details and a brief discussion in Appendix
RELATED WORK
Multivariate Quantiles. Scalar quantile regression estimates conditional quantiles of Y R given X E IR,, typically using linear-in- iaaaiiiiffeatreee models trained via the check loss (Koenker & Bassett 1978 Koenker 2005). Extending this framework to the multivariate setting is challenging due to the absence of a natural order. Early generalizations include spatial quantiles (Chaudhuri (996) and depth-based quantiles (Hallin al.] (222), though these lack transport-mas interpretation. From perspective, multivariate quantiles are defined as optimal transport (OT) maps from a reference distribution, inducing center-outward ranks and quantile regions with desir- able geometric and statistical properties (Chernozhukov et al. 2017 Hallin et al., 2021 Hallin & Konen 2024; del Barrio et al. 2025). The conditional vector quantile function (CVQF) of Carlier
a gecasure-transportaeee
et al. 2 models the quantile map as affine in X and estimates it via variational OT (Carlier
ct al. 2017), with subsequent extensions to nonlinear embeddings (Rosenberg et al. 2023) and non-
parametrie rank estimation (del Barrio et al. (025). However, prior efforts construct continuous VQR models (Vedula et al. 2023b]aa al Sun et al. (2022) have retained the Miine-in-X-embeddee assumption thereby constraining the expressivity of the learned quantile maps. Moreover, these methods do not provide a principled way to estimate continuous rank functions, instead produc- ing discrete, pointwise solutions. Finally, while scalable solvers based on entropic regularization have been developed, to the best of our knowledge, no previous work has scaled VQR using neural optimal transport, proposed in this paper.
Neural Optimal Transport. High-dimensional OT is challenging due to the nonlinear dual formu- lation. One approach employs entropic regularization, enabling Sinkhorn iterations and stochastic gradient solvers (Cuturil
et al. Seguy et al. [0018] Carlier et al. 2022). which scale well but introduce bias that may distort the geometry of quantile maps (Rosenberg et al. 2023). A second approach parameterizes convex potentials with input-convex neural networks (IC- NNs; Amos al. 2017; Makkuva et al., 2020 Amos 2023), ensuring monotonicity and invertibility of the learned map. Conditional potentials (and Monge maps) have been proposed in Bunne et al.
(002)) but are learned in a supervised way (from examples of conditioning and target distributions) and never from unique joint sampling using the framework of Carlirr et al. (2017) as proposed in
our work.
Multivariate Conformal Prediction. Conformal prediction (CP) constructs distribution-free pre- dictive sets with coverage guarantees. In the scalar case, conformalized quantile regression (CQR; Shafer & Vovk, 2008 Romano et al. 2019: Angelopoulos et al. 2023) adjusts quantile estimates to achieve valid intervals. For multivariate responses, naive coordinateiie CP yields conservative rectangles; scalarized scores via norms maxima produce balls or boxes, but remain restrictive. Structured approaches include deep generative embeddings (Feldman et al.] 2023) and copula cali- brations (Messoudi et al. 2021). Dheur et al. (2025) propose conformity scores based generative
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Dataset
EC-NQR
C-NQRg
C-NQRy
AC-NQRg
AC-NQRy
CPF
FN-VQR
VQR
CPQ
CVQR
Star
0.197
0.184
0.184
0.182
0.197
0.247
0.271
0.270
0.274
0.443
Glasses
0.748
0.785
0.812
0.771
0.810
1.657
2.017
1.964
0.931
1.170
Banana
0.111
0.072
0.073
0.073
0.012
0.069
0.398
0.389
0.237
0.401
Cowvex Stor
0.184
0.182
0.191
0.191
0.262
0.261
0.252
0.448
Corvex Glasses
0.650
0.636
0.068
0.637
0.659
0.760
1.951
1.961
0.793
0.933
Convex Banona
0.103
0.101
0.071
0.070
0.070
0.069
0.397
0.391
0.211
0.425
Training
sec.
sec.
sec.
sec.
sec.
Inference time
1.71
1.21 sec.
1.76
1.12 sec.
1.34
Table I: S-W2 between the ground-truth and empirical distributions. We report the per-epoch train- ing time, averaged across all datasets, and the average inference time required to compute the inverse c-transform for 8192 elements. The best result is shown in bold, and the second best is underlined.
models or aggregated p-values.
In contrast with previously developed methods based generative modeling (Feldman al. 2023 Dheur et al., 2025). We propose an explicit generative model, that approximates a canonical quantile function (Hallin et al. 2021).
Very recently, the use of OT-based ranks and quantiles has been exploited in conformal prediction. In two concurrent works, Thurin al. (2025) define conformity scores from discrete OT ranks, while Klein et al. (2025) leverage the same construction albeit with entropy regularized discrete OT. By construction, these two approaches are not adaptive, i.c. the size of the conformal set does not depend on X. Thhrrin et (2025) did propose an adaptive variant based on conditional den- sity estimation with --nnerest neighbors. Our direct learning of neural VQR does not depend on conditional density estimation and we expect it to perform better in high dimensionality settings.
NUMERICAL EXPERIMENTS
6.1 NEURAL OPTIMAL TRANSPORT
To evaluate the generative performance of our models, we conduct extensive experiments. Whenever ground-truth operator is required, we parametrize the datasets using a convex potential function, see Appendix H.2 for details. EC-NQR, C-NQRu, C-NQRy, AC-NQRu, AC-NQRy are the meth-
ods described Section 3 We measure the generative performance against FN VQR (Rosenberg et al., 2023), VQR (Carlier et al. 2017). CPF (Huang et al. 2021), CVQR (Vedula al. 2023a and
CPQ (un al. 2022).
Metrics. We employ three categories of metrics: (i) Wasserstein-2 (W2) and Sliced Wasserstein-2 (S-W2) distances; (ii) Kernel Density Estimate & distance (KDE-L1) and Kernel Density Estimate Kullback-Leible divergence (KDE-KL); and (iii) Percentage of Unexplained Variance (L2-UV: rotin et al. (021). Metrics (i) and (ii) quantify the fidelity of the learned distribution to the target density, while (iii) assesses the extent to which the ground-trute quantile is recovered. Additional implementation details are provided in Appendix H.l
We evaluate three synthetic datasets originally introduced in the discrete setting of quantile regression (Rosenberg et al. 2023): Banana, a parabola-shan shaped distribution whose curva- ture varies with latent random variable; Star. strene-p star whose orientation is governed by a latent variable; and Glasses, a bimodal distribution with sinusoidally shifting modes. We denote convex-potential counterparts Convex Banana, Convex Star, and Conver Glasses.
Datasets.
Results. The experiments in Table evaluate the generative capability of the proposed neural quan- tile regression framework.
We evaluate the ability our method to recover the underlying quantile mapping. We report the L2- UV metric in Table 2 evaluated on Convex Banana, Convex Star, and Convex Glasses. The results in Table show that the proposed models achieve high precision in reconstructing the true quantile operator.
The experiments confirm the overall performance of the proposed quantile construction, making it suitable tool for subsequent conformal prediction procedures. Extended experimental results
8

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Function
Dataset
EC-NQR
C-NQRø
C-NQRy
AC-NQRØ
AC-NQRy
CPF
Convex Star
1.331
0.211
0.286
0.264
0.425
0.447
Qyix
Convex Glasses
0.348
0.332
0.068
0.203
0.109
2.268
Convex Banana
3.942
3.784
0.212
0.106
0.206
9.479
Convex Star
2.746
0.360
0.351
0.393
0.525
0.267
Quix
Convex Glasses
0.678
0.535
0.732
0.985
1.096
1.726
Convex Banana 9.400
7.665
0.660
0.545
0.569
16.537
Table 2: 1.2UVV of the true quantile function, evaluated on generative processes parameterized by convex potential networks. The best result is shown in bold, and the second best is underlined.
01-3P
01-1P-
ELL
PB
RUP
IBS
RPBS
sce20d
bio
blog
2.0
1.5
0.2
1.0
0ng
0.0
0.5
0.0
-0.2
0.0
-0.2
-0.4
0.6
10
-0.8
Figure 1: Log-volume of the prediction sets, normalized dy. of the resulting prediction for different methods. Results averaged over 10 independent data splits. Nominal miscoverage level 0.1
including experiments on scalability, visualisation of learned generative model and full list of metrics studied datasets available in Appendix H.3
6.2 CONFORMAL PREDICTION EXPERIMENTS
We further evaluate conformal prediction by constructing prediction sets on real-world datasets using the methods described in Section Extended results are presented in Appendix H
Methods. We use AC-NQRu the base model to implement our two conformal methods: PB(Crb) and RPB (CTP)), In addition to fitting vector quantile regression model directly y. we also fit both methods signed residuals y f(x). where f is Random Forest regressor fit on 25% of the training data (PBS and RPBS in the plots). We consider OT-CP and OT-CP+ (Thurin et al, 2025). well the local Ellipsoid method (Messoudi et al.) 2022 for comparison.
Metries. We evaluate performance using three metrics: (i) marginal coverage, (ii) worst-slab cover- age (Cauchois et al. 2021) and (iii) average prediction set volume.
Datasets. We evaluate on standard multi- multi-target regression benchmarks used in previous work on
uncertainty estimation (Plassier et al. 2025 Dheur et al., 2025): scm2 0c, sgemm, blog, and bio. For the single-target datasets blog and bio, we follow Feldman et al. (2023) and add one of the features second output. The resulting response dimensions are 16, 4, 2 and 2, respectively. We use preprocessing procedure of (Grinsztajn et al. 2022).
Discussion. PB and PBS provide competitive conditional coverage and smallest volume at the same time on three out of four datasets. The re-ranking step of RPB and RPBS allows to achieve slightly sharper conditional coverage, but the increase in prediction trade-off. Overall, shows that for our quantile regression models the split conformal calibration is enough. Our methods provide scalable training enable building competitive conformal predictors.
volume make it questionable

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OT CP
OT OP+
E11
P5
RPB
PSS
RPB5
scr20d
bio blog
0.55
0.30
Woret cab couerge
0.85
0.30
0.70
Figure 2: Worst-slab coverage different methods. Results averaged over independent data
splits. Nominal miscoverage level a 0.1
7 CONCLUSION
We introduced framework for multivariate conformal prediction based on convex potentials and op- timal transport. Our approach leverages neural quantile regression with input convex neural network parameterization to construct valid and efficient prediction sets. Through experiments synthetic benchmarks and real-world multi-target regression datasets, we demonstrated strong performance in terms of coverage and set size, while maintaining scalability in higher dimensions. Comparisons with existing baselines further highlight the robustness and flexibility of our method. Future work includes extending the framework to broader classes of generative models and exploring tighter efficiency guarantees in high-dimensional regimes.
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USAGE OF LARGE LANGUAGE MODELS (LLMS)
LLMs were used as a general-purpose assistive tool during the preparation of this paper. Their us- age fell into two categories: (i) for writing assistance, they helped improve clarity and readability of certain passages through language refinement and (ii) for coding assistance, where they provided support with code completion and debugging. LLMs were not used for research ideation, experimen- tal design, theoretical development, or analysis of results. All substantive contributions, including the conception of ideas, methodology, and experiments, were made by the authors.
REPRODUCIBILITY STATEMENT
We provide the full code to reproduce our experiments supplementary material and will release it publicly upon acceptance. All experiments were conducted on publicly available datasets or datasets created ourselves, which will be released alongside the code. We ran experiments with multiple seeds, if applicable, and report summary statistics.
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A EXTENDED STATE OF THE ART
From scalar to vector quantiles. Classical quantile regression (QR) estimates conditional quan- tiles of scalar response Y E R given features X E IP, providing a flexible alternative to least squares for modeling heterogeneous effects (Koenker & Bassett 1978; Koenker (2005). For quantile level u E (0, 1) and feature map sp(x), a standard linear QR model assumes Qy (u
a) = 3(u) p(x),, with 3(u) obtained by minimizing the check-loss. While univariate QR theory well-developed, extending these notions multivariate response Y E R is challenging due to the lack of natural total order on IRd. Many generalizations have been proposed, including directional or projection quantiles (reducing to scalar quantiles along particular directions) and ge- ometric or spatial quantiles (e.g. Chaudhuri (196). as well as definitions based on statistical depth (e.g. Tukey's ss halfspace depth) that yield central regions viewed as multivariate "quantiles." " How- ever, these early notions only partially extend scalar quantile properties and generally do not yield a unique quantile mapping for Y. A recent breakthrough comes from the measure transportation perspective, which defines multivariate quantiles as the optimal transport map pushing reference distribution (usually the spherical uniform, or uniform on the unit hypercube) onto the distribution of Y. This approach rooted in Brenier's theorem monotone optimal transport maps (Brenier) 1991) yields bell-behaved center-outward distribution and quantile functions that assign each point in Rd multivariate rank and sign with distribution-free properties. The resulting quantile regions are nested, have correct probability contents, and enjoy equivariance properties generalizing the one- dimensional case. These concepts, introduced by [hhrnozhukov et al.) (2017) and further developed by Hallin et al. (2021). provide a rigorous multivariate analog of the quantile function; see (Hallin & Siman 2017) for a survey of earlier definitions. Recent work continues to refine this frame- work: Hallin & Konen 2024) compare geometric vs. transport-based contours, and nonparametric oultip quantile regression methods based on center-outwand ranks have been proposed (del
Barrio et al. 2025).
Vector quantile regression (VQR). Building on optimal transport ideas, Carlier et al. (2016) in- troduced the conditional vector quantile function (CVQF) Qv]x(u, x) for Y E R". This is defned as a (a.e.) monotone map in u - specifically, the gradient of a convex function in the u argu- ment- such that for each fixed x. Qy)x((,z) pushes the uniform distribution on [0, 1j4 forward to the conditional distribution Y X = x. In analogy to the scalar case, one can represent Y as Y = Qr)x(U; (U, X) with U Unif([0, 11d) independent of X. This generalizes the scalar quan- tile relationship Y Qy QU, X) for U Unif(0, 1), providing a powerful characterization of the conditional law of Y by a deterministic map on the unit hypercube. In practice, VQR imposes a parametric form on the CVQF; for example, the original proposal assumes an affine structure Qyix(u,2)) x) = a(u) + B(u) T x (with ((x) E IR and B(u) E Rdxp) and estimates these functions by solving a large-scale optimal transport problem under empirical data constraints. The solution can be found via convex dual formulation analogous to Koenker's 's linear program, ensuring the fitted Qyix is monotone in u (i.e. cyclically monotonic) (Carlier et al. 2016 2017). This yielded the first notion of "quantile regression for vectors," " including strong theoretical guarantees on con- sistency and uniqueness under appropriate conditions. Since then, a number of extensions have been proposed: Rosenberg et al. (2023) introduce a fast nonlinear VQR model (e.g. using kernel neural network features) while preserving monotonicity, Vedula et al. (003b) develop a continuous VQR formulation that treats in continuum (rather than on finite grid of quantile levels), and fully nonparametric approaches based on center-outward quantile functions have appeared (del Barrio al. 2025). Each of these methods seeks to balance flexibility and computational tractability while maintaining the defining property that u + Qrjx (u, is gradient map (hence invertible and order- prder-preserving in the multivariate sense).
Computation. Implementing VQR at scale poses significant challenges. The initial algorithms of Carller et al. (2016) and Carlier et al. (2017) relied on discretizing the unit hypercube [0, 1]d (for a set of representative u values) and solving a large linear program, which becomes computationally expensive as d the number of quantile levels grows. Two recent strategies have substantially improved the scalability of VQR. First, Carlier et al.] (2022) propose an entropic regularization of the OT problem. which smooths the objective and leads to differentiable dual formulation. By applying Sinkhorn-type iterations or gradient- based optimization on the regularized dual, one can efficiently approximate the CVQF without solving huge IP,, even for continuous u spaces.
15

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This regularized VQR approach yields an accuracy coouraccccommuuatiinni cost trade-off controlled by the entropy penalty, and it has demonstrated orders- of of-magnitude speedups on problems.
The second approach uses deep learning to represent the convex potential of the CVQF: Makkuva et al. (2020) propose to parameterize Qy xu, 2²) Vuth(u, x)) where di is input-convex neural network in u. By training v on data (using a suitable loss derived from the OT characterization), one obtains a VQR model that can handle high-dimensional X and Y and large sample sizes. This method, part of a broader trend of using neural networks for OT map estimation, sidesteps explicit discretization by leveraging automatic differentiation to enforce convexity u. Both the entropic- OT and ICNN-based approaches have made it feasible to learn multivariate quantile functions on modern datasets, a task once thought impractical (Huang et al., 2021 Kan al. 2022). For addi- tional background on scalable optimal transport techniques that underlie these advances, see (Peyné et al. 2019).
Conformal prediction. Conformal prediction (CP) provides distribution-free predictive uncer- tainty sets with saitle coverage guarantees. In the scalar Y case, common to combine quantile regression with conformal calibration. For example, conformalized quantile regression (CQR) uses holdout data to adjust the initially estimated interval [Ωr (a/22 xr), Q a/2 x)] so that it achieves the target coverage 1-a marginally. CQR and related methods yield prediction intervals that are adaptive (varying with x) while retaining rigorous coverage guarantees (Romano et al. 2019: Angelopoulos et al. 2023). However, extending CP to multivariate outputs has proven more complex. Naively applying conformal methods each component of Y .Y,,... ... Ya) and taking a Cartesian product of marginal intervals yields a rectangular prediction region that is valid but often overly conservative (covering significantly more than 1 - a of the probability). More refined strategies have been proposed to account for dependence between coordinates. One line of work defines a scalar nonconformity score from the multi-output residual, for instance using a norm prred Ynuell or the maximum deviation across coordinates; this yields prediction balls boxes aligned to the chosen norm. While simple, such choices typically lead to symmetric or axis-aligned regions that may be suboptimal in shape and volume. For example, the PCP method of Wang et al.) (2023) leverages an implicit generative model to draw random samples from Y X = I and con- structs the prediction set as a union of Euclidean balls (of a fixed radius) centered those samples. This approach guarantees marginal coverage and can improve sharpness over naive intervals, but us- ing a global radius for all can lead to over-coverage in low-variability regions and covercc in high-variability regions. Alternatively, some works shape the prediction set as an ellipsoid by incorporating covariance structure: e.g. using a single global covariance estimate (Johnstone & Cox 202T) or a local covariance around x (Messoudi et al., (022) to define a Mahalanobis conformity score. Such ellipsoidal regions capture linear correlations in Y and are typically smaller than axis-aligned boxes, but they still assume an (approximately) elliptical and unimodal error dis- tribution, which may be inappropriate for complex multimodal targets.
Another class of methods seeks to learn joint representation or dependency model for Y before applying conformal. For example, Feldmann et al. (2023) train a deep generative model to embed Y into a lower-dimensional (ideally unimodal) latent space and perform conformal quantile regression in that space, producing flexible regions when mapped back to IRd.. Similarly, Messoudi et al. (2021)
and subsequent works leverage copula transformations: they calibrate marginal predictive intervals at miscoverage levels chosen to optimize the volume of the resulting joint region, effectively shaping the prediction set according the dependence structure of Y. In particular, Zhang et al. (2023) extend copula- based conformal prediction by allowing different significance levels for each output dimension and directly optimizing the hyperrectangle volume under the coverage constraint. [Sun] & Yu 2024) provide theoretical analysis of such copula-shaped prediction sets, proving that the empirical copula approach achieves samiee validity under i.i.d. assumptions. These methods produce tighter joint regions than the naive Cartesian product by allocating miscoverage intelligently across coordinates, though they often rely on either simple parametric copulas or numerical search to balance the marginal intervals.
Very recently, Dheur et al. 2025) conducted a comprehensive study of multi-output conformal meth- ods, proposing in particular two new families of conformity scores. One uses generative model (e.g. an invertible normalizing flow) to transform Y into a space where conventional CP can be applied moordinate-wise and the other defines computationally efficient scalar score by combin-
16

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ing coordinate -wss conformal p-values (essentially summing their logarithms). Both approaches attain finite-sample marginal coverage and offer improvements in conditional coverage. Notably, conceptually similar idea was introduced concurrently by Fang et al. (2025), who also leverage normalizing flows to define nonconformity in the latent space. Their method (CONTRA) maps high-density regions in the latent space complex but high-coverage regions in output space, yield- ing von-axis-aligned prediction sets that outperform standard hyperrectangles or ellipsoids. Despite these advances, none of the above techniques exploits the full geometric structure of multivariate quantiles or ranks. This gap has been filled by two concurrent works that integrate the measure- transport perspective into conformal inference.
Thurin et al. (2025) introduce OT-CP, which uses the center-outward rank function of Hallin et al. (2021) to define multivariate order statistics. In essence, they compute the "rank" of a test point y among past observations in I2 via the empirical center-outward distribution (obtained by optimal transport), and use the corresponding multivariate quantile level as the nonconformity score. This yields a prediction region for new X z by including all y whose center-outward rank is above a certain quantile (determined by the calibration set)-intutivvll, the set of points that lie among the (1 --aa a)) fraction most central (least outlying) under the conditional distribution of Y X x. Independently, Klein et al.] (2025) develop a related approach that also relies on optimal transport to order multivariate outputs. They formalize the notion of distribution-free multivariate quantile regions and provide finite-sample coverage guarantees for both exact and approximate transport maps. These OT-based conformal methods leverage the geometry of Brenier maps (i.e. conditional Monge-Ampère transports) to construct flexible, data-dependent prediction sets in 18 that adapt to the local distribution of Y X x. By exploiting the vector-quantile structure, they can achieve tighter coverage with complex (even non-convex) regions while still guaranteeing the rigorous cov- erage properties that make conformal prediction attractive. However, the use of optimal transport maps can be computationally expensive in high dimensions, and in practice one might need to trade off some statistical efficiency for tractability when estimating the transport.
Finally, an alternative direction is to explicitly optimize prediction set volume subject to cover- age, rather than relying on a fixed conformity score. Braun et al. (2025) propose an optimization- driven framework that learns minimum-volume approach, the predictive model is trained jointly with a parametric prediction set (for example, an adaptive norm-ball whose radius may vary with x) to minimize the volume of the set while enfore- ing coverage on the training data via a surrogate loss. This procedure effectively learns the shape of the prediction region that best captures specified proportion of the data. By conformalizing the learned region (i.e. slightly expanding it to guarantee - a coverage on a holdout set), the method yields valid prediction sets that are much tighter than those from standard split-conformal methods. Such approaches highlight an exciting trend of combining machine learning and conformal infer- ence: rather than treating the prediction algorithm as a black box, one can optimize the model and its uncertainty quantification in tandem to achieve improved efficiency (smaller, more informative prediction sets) without sacrificing the finite-sample guarantees of CP.
covering sets for multivariate regression. In their
B REMARK ON CONTINUITY OF DISTRIBUTION FUNCTION
The requirement that Fyjx=z admit density in Assumption is stronger than needed, see (Ghosal & Sen 2022), and is not customary for conditional vector ranks. For our results, it suffices that the source (reference) distribution Fy be absolutely continuous with finite second moment (e.g., uni- form on [0, 1]dd or Gaussian). No density assumption on Fy is required; in particular, Fyxx== may be discrete.
Under these conditions, for each there exists convex potential fa such that the Brenier map Qx)x(, = Vip, is defined Fy-a.e, is unique Fy-a.e, and pushes Fy forward to the conditional law:
(Qyxx(;z)) Fu = Fyix
The (conditional) vector rank is given by the gradient of the convex conjugate, Qy Qylx( (,,2) = V which is defined Frix= (when FyJx= is discrete, interpret Qylx as any measurable selec- tion from O.p)). It transports back to the reference:
(r (u, x),
for Fy-a.e u.
17

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Consequently, could work under the standing condition that Fu is absolutely continuous with finite second moment; Theorem holds verbatim with the above interpretation.
C ENTROPY-REGULARZER NEURAL VQR
Let X, be Polish spaces with Borel g-algebras, and let m be the marginal law of X, v(dx, dy) m(dx) (dy) the joint law of (X, Y), and u(dr, du) m(dx) ([du) the joint law of (X.U) U' (where ş is the marginal distribution of U). For 0, the entropic-regularized conditional OT problem reads (Carlier et al. 2022)
{ { ydv + KL (1 s.t. =
min PEM, (Xxxxxt)
(10)
This is strictly convex problem with linear marginal constraints; KL denotes the Kullback-Leible divergence. (10) specializes the standard entropic OT to the conditional setting by constraining the two (X, marginals of
Dual formulation via Renchel-Rockafellar: We introduce the dual potentials v; X' x ) + R and sp: X xu R. A direct application of Fenchel-Rockafeee duality yields the (unconstrained) dual
vly x) v(dx, dy) p(u,a) p(dx, du) inf 0..0
Hx
term Xx
ʃ exp u (''=-p(u,) v(de, dy)ā (11)
+
with zero duality gap and attainment under mild assumptions (tightness and finite entropy) The inequality constraint of the unregularized dual is absorbed by the exponential term barrier in (IT) This could be solved using purely stochastic optimization with NN parameterization of the two dual potentials (yy, x)) and ypu,, 2) similarly what was proposed in (Genevay et al. 2016 Seguy al., 2018). But from practical perspective the exponential in the loss is particularly hard to optimize with numerical stability problems. This is why [eenvaa et al. (2016) proposed to remove one the potentials using the smooth version of the c-conjugacy detailed below.
KKT//irst-order conditions: soft c-conjugacy. Assuming v and j admit densities w.r.t the Lebesgue measure and differentiating the dual objective in II w.r.t. tb and ip gives the optimal- ity (stationarity) conditions
the (y, .) = =log
exp v-ge(um) p(du),
(12)
(u((u,)) = log
exp
u (-
Vnr (yl()))
(13)
which are the entropic ("soft") c-transforms, i.e., log- log-partition functions of exponential families
induced by the bilinear cost c(u, y) -u y. At E 4 O the identities (12) 10-(1) T c-converge the
hard Fenchel conjugacy &b _p*, recovering the unregularized dual feasibility (u,, + (1y, 2
y with equality on the support of the optimal plan.
Reduction to a single potential (semi-dual). Eliminating & in 11 via 12) yields an equivalent unconstrained problem in yo:
Uc(7)) = Ex [o(U) + E( log
exp [TTY (TT-p(u))) plddu)
(14)
which is precisely the regularized analogue of the conjugate-based loss in the unregularized case (log-sum-exp replaces the sup). This problem very interesting from an optimization perspective because now unique dual potential needs be optimized and the log- lum-sup-exp can be implemented in a much more stable way than the exponential in the dual (TT) But then the inner expectation in the right part of (14) cannot be computed exactly, which discuss next.
18

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Gibbs conditionals and gradients. Define the Gibbs conditional density (a.k.a. Schrödinger bridge "posterior")
Tp y, x)) exp
'x" ""y-g(uaa) jī(du).
As in the not regularized case, parameterize the potential ( with neural network. We denote by f the parameters (weights) of this network. Using the log-partition derivative identity, we get that Va ((pe) admits the "positive minus negative phase" form
Volle(po) ~ (opp((XU)) ) R (+Y.X) Vo ()))
(15)
obtained by differentiating the log pee--pattiionn in (14). In practice, the inner expectation is estimated by Monte Carlo with U drawn either from Kypa Y, X)X) or via importance sampling from j with the usual exponential weights.
Quantile and rank maps under entropic regularization. If + (u, x.) is (strongly) convex and smooth, the regularized analogues of the conditional vector quantile and rank are
Q (u, x) == V (u,
(16)
(o (v,x) (g. = (y EUNT (lu.z)) (U),
(17)
where the last identity follows by differentiating (22 Equations (16 17) are the entropic counter- parts of the unregularized identities and reduce to them 4 O.
Limit 4 O. As e 0, e log exp(())/e) - sup(-), so
Uc(p)) - E. ( U)] + E. 210
recovering the unregularized loss with the hard Fenchel conjugate and the transition from the strained dual (inequality) to the unconstrained conjugate form. In the same limit, Tp(- y, x) concentrates on the (possibly set-valued) argmax of u + u Tu ppu,,), and 16 the OT maps of the unregularized problem.
converge to
D CONDITIONAL CONVEX POTENTIAL FLOWS
Conditional (partially convex) construction. transport by partially input-convex potential
Given covariates X,, model the conditional
: R x
X + R,
(+ ++((((; convex (strongly convex) for each fixed æ,
and define the conditional convex potential flow (a.k.a. partially convex potential flow)
Qvxx(u.x) = [u)(u; U~ fo Y Y via Y (U,
Under absolute continuity of fy æ) (see Assumption 2 the conditional rank map (inverse quantile) exists and is the gradient of the conjugate:
(v, V, (y:
and the two maps are inverses a.e. (in and y) for each v. For any (u, a) such that the inverse is well defined.
fri x)) fo (Qr x)) det ()
(18)
Equivalently, writing y = Vup(wx)) with 1 (V.
log ( x)) = log fu (u) log det 222 ((;
Thus maximum likelihood amounts to estimating oo so to match the pullback Qy1 XY, to the prior fu, while penalizing the local volume change through the (log) Hessian determinant. In practice, the deg-determin and its gradients can be computed with Hessian products, using stochastic Lanczos/trace estimators and conjugate-gradee solves, yielding unbiased ((1)-memory estimators that scale high dimension.
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Inversion and sampling.
For any (y,z), inversion is convex program:
Qvix x) arg min (pl;;)) y u,
weRd
whose optimality condition Vui(( (u;((u;) x)) recovers the required This is precisely the evaluation of V Vy x:) and can be carried out with off-the-shelf smooth convex solvers; batched inversions reduce to minimizing summed potentials over independent inputs.
Under mild regularity (convex support and densities), there exists a measurable conditional vector quantile Q that is the gradient (in of convex potential and pushes U to Y X x;; the inverse rank is the gradient (in y) of the conjugate, and Qr solves the W2 OT problem condition- ally on z. Hence the partially convex potential flow inherits both identifiability (a.e. uniqueness) and optimality properties in the conditional setting.
instantiate vp(:;)) with partially input-convex networks (e.g.. PICNN/PISCNN) to guarantee convexity in while conditioning on z, and add quadratic |||ul| when strong convexity is desired. Universality of ICNNs in approximating convex functions then lifts to distributional universality of the induced conditional flows and convergence to the conditional OT maps.
Parameterization.
E NUMERICAL IMPLEMENTATION
This section details architectures, solvers, and training procedures for our neural vector quantile regression (VQR) models, both in the unregularized and entropic-regularized settings, together with the amortized conjugate prediction used to accelerate training. We emphasize implementation choices that preserve oonoeoity/moooooniiity and lead stable gradients, and we provide concrete defaults for reproducibility.
Notation recap. We parameterize conditional convex potential ypy: X that is convex in E U C Rdd, for each fixed x E X. The conditional vector quantile and rank maps are the gradients of pp and its Fenchel conjugate vp (see Section 2):
Qrix [u, x) = V (u,
Q (r.x = (v.
The conjugate evaluation at (y. x) solves (g x) E arg max { so (u, x)}. By Danskin's 's theorem, gradients w.r.t. 0 do not require differentiating through úg: only Voyo üg is needed.
E. PARTIALLY INPUT CONVEX NEURAL NETWORKS (PICNN)
We instantiate Pe as a Partially Input Convex Neural Networks (PICNNs; Amos et al. 2017) that is convex and conditions on x:
(u,x) + x))(u.x) PICNNNu,,;;)
with layerwise updates
G+1 = ğ: (W,c, be),
2,+1 = gi ( (z: a + W:!") [u (W("o)
a; + [b")]] + w!
ci + 0z)
and output pp x) = zk. We initialize co = x, 20 = Here denotes the element- wise product. We enforce elementwise nonnegativity of w((x) and [+]+ via a Softplus reparameterization:
w -log exp ( (w(()))
w R
(19)
[W! 0 ((2)) log exp
W (W(2c) 4
)(+))
(20)
We convex, non-decreasing activations for 9s. 9s. which guarantees convexity in while retain- ing expressive power. We optionally add a quadratic term & | (trainable a 2 0) to obtain strong convexity, improving stability of the inner argmax (Amos et al. 2017 Proposition 2). We choose
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Softplus non-linearity for 9, and ELU non non-linearity for C₁. Following Huang et al. (2021) we utilize activation normalization ActNorm layers (Kingma Dhariwal 2018) before applying the 9: non-linearity. Final architecture of one iterate hence becomes.
G+1 = ELU(W;c +b,),
4+1 = Softplus (ActNorm (x [W/tol [,,
+ W (u [W|ut] + +1 + w 4 + +bb)):
Practical tips (PICNN).
(i) Normalize u and y scales (c.g. standardization) to ease optimization;
(ii) We use weight decay on 0 and (if enabled) small ridge avoid flat directions;
(iii) We clip gradients of yo to bound the Lipschitz constant of u + v.po(u,
E.2 PARTIALLY INPUT STRONGLY CONVEX NEURAL NETWORK (PISCNN)
(I PICNN
which strongly convex in and yields strictly concave inner objective u + uuTy po x)), ensuring unique maximizer (g 2) and faster, more reliable inner solves. We treat a as positive scalar parametrized by e. where w is a trainable parameter. In all our implementations, enabling eliminated numerical non-uniqueness in the conjugate and reduced inner iterations.
E.3
COMPUTING THE CONJUGATE: INNER MAXIMIZATION
Given (y, x) and current 0, we compute
To (y, x) E arg max Jo(u;y, Jo(u; uTy up)oo(u,x). well
Gradient and Hessian. VoJo(u; 2) = = y Vuipo 2) and v2 Jo g, -V (u,.)o( 0. With PISCNN, v (o x) ? a ensures strong concavity.
Solver. We minimize Jo with L-BFGS. For stability:
1. Warm start. We initialize the solver from amortized predictor uo x) when available (see Section EE.4) or otherwise initialize it at u Fu.
2. Domain handling. I is ball/hypercube, we project the solution after each step: u <- (u). Stopping. Terminate when Joll I m || (u: y. 2) Jo (7/411111 x) S Eabj or after
Kmax steps (defaults: Enorm 10 Eobj 10 Kmax 1000).
E.4 AMORTIZED CONJUGATE PREDICTION
To avoid expensive inner solves at every iteration, we learn differentiable predictor ūe ) x x- that approximates ü. (y, x) and serves as a warm start for L-BFGS solver. We parametrize (=().)) as feed forward neural network with residual skip connection to encourage identity at initialization
2) MLP
+ Woy + by.
Training losses. Several loss functions have been explored in the literature. Objective- based losses (Dam et al.) 2019 Amos 2023) optimize the network to predict the maximum Alternatively, one may V. .:) (== & via the residual loss Vie =
of the conjugate by maximizing Vony
enforce the first-order condition
E(y,x)~xy> x |I| Vuseo (u, x))==ū)(y.)) vlll). If the true conjugate ( (v) (5) is available. one regress directly with Va E(y,x)-Fyx [l] (1), u (v) In practice, observe no major differences between these approaches and therefore adopt V2 as our loss choice (see Algorithm
21

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E.5
ENTROPIC-REGULAAIE SEMI-DUAL
When using the entropic semi-dual Ug((p) (see Appendix C log-sum-exp:
we replace the hard conjugate with a
U po(U,X))
Uz(pp)) (x,,) E(x,))
llg Bu exp
E
Monte Carlo and stability. We approximate the inner expectation with ii.d. samples Uj Fr. using a numerically stable log-sum-exp with 64-bit accumulation. We found m E [512, 1024] adequate benchmarks, and re-sample the U, each iteration. In the J. 0 limit, this recovers the unregularized loss. We intentionally set high amount of samples for dual objective estimation to avoid effects related to high bias of logsumexp estimator.
Gradients. The gradient has a positive-minn--neeatin phase form using the Gibbs weights (see Appendix C and equation [15),, which implement without storing the full batch xm tensor by streaming accumulation.
E.6 TRAINING LOOPS AND ALGORITIMMS
We describe three loops: (i) Neural Vector Quantile Regression without amortization Algorithm
(ii) Amortized Vector Quantile Regression Algorithm and (iii) Entropic Semi-dual Algorithm
All use Adamli (initial LR of 10-2, weight decay 10 with cosine annealing (LR decaying to 0), batch size 1024, and gradient clipping at 10. We sample U Fr standard Gaussian unless otherwise noted. See Appendix H.2 for dataset-specific details. We use warm restarts for amortized network, restarting the learning rate 10 each 10 epochs.
Algorithm Neural Vector Quantile Regression Training (C-NQR)
Input: dataset {TT,, y4) PICNN (pg: -R
2: Sample mini-batch B C (1, nn)
3: Initialize V.+-
4: for each i B do
5: ug + maxwex J (u,
Run L-BFGS for each y/s starting
6: (0) + Jp((sa.)) (ū,, Vi)
7:
Sample ~ (a,
8:
(0) - (pl(
9:
(0) -
end for
11: Compute V [8] (0)
> Do not propagate gradients through
12: Update 9 with Adam
E.7 CONFORMAL. METHODS IMPLEMENTATION
Here, provide a detailed description of our implementation of the methods introduced in Sec- tion For all proposed approaches, we start with an estimate Qyix (x) that we obtain using training set Dtrain- All conformal methods operate on a separate held-out calibration set Deat- Since we need to replicate our uncertainty estimation experiments for multiple splits and datasets, we use the Amortized Neural Vector Quantile Regression version of our algorithm.
Split Conformal Prediction with Monge-Kantorovin ranks. An instance of classical split con- formal prediction using a score derived from our vector quantile regressor.
Fixed re-ranking. To account for the misspecification of Q (v. x) we introduce an intermediate re-ranking of U.. We follow the approach of Thurin et al. (2025). but instead of a separate base model, we directly use our estimate: S, U, = Q (Y,, X,) E gdd We divide our calibration set into parts: the first part is used to estimate OT map R: 2 Lt' and the second part is
22

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Algorithm 2 Amortized Neural Vector Quantile Regression Training (AC-NQR)
1: Input: dataset {Zz, 3/4 (+)]=11 PICNN :po xuxxxa X + RR z)( y x - +u
2: Sample mini-batch B C (l,,.... [I,...,
3: Initialize Vi2 <- 0, Va +
4: for each i B do
5: ü; + (z 2;)
6: 01 arg max J (po(-22)(,,1))
Run I.-BFGS for each ys starting 0
7:
(6) t (p(( (ū,, Vi)
8:
Sample u4 ~ *N(00
9:
(0) :pluu, xii
10:
(0) - (0) + (1( (.)
11:
Va(0) < Va(0) + l|ūq 0.112
12: end for
13: Compute V [ (0)) and V (.
Do not propagate gradients through
14: Update 0 and 1
Algorithm 3 Entropic semi-dual training (EC-NQR)
1: Input: dataset ((r;, y/ y:) J= PICNN yp: U x X + +R
Sample mini-batch B (I....,,)
3: initialize C.p + 0
4: Sample i.i.d. Uij Fu
5: for each i E B do
-o
6:
(0) e log 2ラ=1 exp
> E [10-3 11003,10-1
7: Sample u, ~ Fu
8: (.) pa (a Iz);
9: (0) (0) + (0) (0)
10: end for
11: Compute V (8 C (0)
12: Update 0 with Adam
Algorithm Pull-back split conformal prediction
1: Input: dataset Deal ((X,, M) trained model Qy 1x (v. x),, new test point (X Yeest)
and the desired nominal miscoverage level
2: for each i E {1, (1, do
3: U, - X X,)
4: S ||A|||
5: end for
6: PI-a < [nn + 1)(1 - a) largest S₁
7: Čpb (Xtest) t y ; |1yrix (v. Xteat) S PI-aj
used conformalize the result. In our experiments, follow the original authors' approach and use 24' U(S - uniform distribution on the unit ball. To evaluate the map R. on the new point, map it to the corresponding closest point from the first calibration part.
We use the code of Thurin et al. (2025) to estimate R (we divide the original calibration set into two equal parts). This implementation uses the renowned POT library (Flamary et al. 2021), which provides efficient implementations of the various optimal transport techniques.
23

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Algorithm 5 Re-ranked pull-back split conformal prediction
1: Input: dataset ral {(X,, K.) trained model Q (1, (Xuest Ylest) and the desired nominal miscoverage level
test point
2: for each i {1, m1,...,1)):..... do
3: Uj U Q (X,
4: end for
5: Estimate R using sample (1U (2))))
(U0 reference sample from L!'
6: for each j E [1,...,ma] do
+ || (xxx))
7: Sj
8: end for
9: PI f (1 + 1)(1 -a) largest Sj
10: Crpb(Xeest) t y: R @rix (U, Xtest) sp--a
Table 3: Model hyperparameters for different datasets.
Dataset(s)
Layer width
Layer depth
Batch size
bio
12
4
512
blog
16
4
512
sgenun scm20d
46
4
8192
10
I
2048
Banana, Convex Banana, Star, Convex Star Glasses, Convex Glasses, Funnel
18
8
256
18
8
256
E.8 HYPERPARAMETERS AND DEFAULT CONFIGURATION
Network sizes. We typically around 10% of available data parameters scale. See Appendix E.8 for details.
Optimization. AdamW (LR 10-2, weight decay 10-4). We use cosine warm restart for amortization network every 5k-10k steps; We clip gradients 1.0.
Inner solver. L-BFGS with Wolfe line search, Kmax = 50 (amortized) or 100 (no amorti- zation); tolerance 10-5; domain projection when U is bounded.
Amortizer. Amortization network copies the potential network architecture in our ex- periments.
Entropic. In all experiments we fix 0.001; 1024 Monte Carlo samples per (I,
These defaults matched the settings used across Section 6.1
6.1 and Section 6.2 (metrics and datasets).
F DEFERRED CONTENT FOR CONFORMAL PREDICTION
We now proceed to provide the deferred content from Section 4 We start by restating Theorem and its proof. Then, we showcase setting where the assumptions Theorem met. Finally, we explain how the OT maps Qyix and Qrix may be used to construct conformal sets using density estimation.
Theorem 4 (Volume-ove -otimmallyyy of pullback balls under radiality). Fix X and reference dis- tribution (u) = ([[w[]) for continuous Φ: [0, 20) - (0, oc) on Ut under the assumptions of Theorem let Qyix and Qyix be the vector quantile and multivariate rank functions. Assume that there exists ja such that for all in the support of Fvix. it holds
del V v, = jz
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and the function > ((r) (r( is strictly decreasing. Let Ta 0 be the unique radius satisfying u(Br,) = 1- a. where it is the law corresponding to Fe and B, = {u: |ulll S r). Define the pullback ball (x = y: (v, S ra Then, (x) minimizes Lebesgue volume among all sets with x-conditional coverage of at least 1 - ax for every measurable A C Ya with P(Y E A X = x 2 a, Vol(C (z)) I Vol(A).
Proof. Let Sz(+) = Qrix() (-). Then, by the change of variables formula for densities:
fvi x) = fu (V)( det vv,,,,v)]
Using the assumption that (u) = <([]u])) and det [v)s,vv)] jz (v)l) Using Carlier et al. 2016 Corollary 2.1), we note that S is Ca and the derivative of a convex function. Thus, holds that y del [V vw)) is positive and continuous, which allow for dropping absolute value to recover
()) = & (vIS) is (1))) =: h.r (1S).
As both o and + ju (x)), are continuous, h2 is strictly decreasing continuous invertible function. Hence, frix (x)x(()))) 2) is a non-increasing function of the U-radius (v)ll and its superlevel sets are pullbacks of Euclidean balls: for each / there exists r(1) 2 0 such that
ty: fx (v, x) 2 8) ty: he (v)l)) 2 8 {v: [|a( s r())-
We first record the probability identity. For any Borel A C Vv..
P(Y A X x) p (1Sx())lv E A)).
Therefore P(Y E [x) 1 (...) a
For volume optimality, note that since Srix (u, hr (() () with hg non-increasing, ery HPD superlevel set {y: (y, 2 t} is (almost surely) pullback set of the form (x)Il < r())). Choosing la so that P(Y E UUix((,xx 2 ta) X x) = - forces 1 (BB((.))
a, hence r(t..) Ta and the HPD set equals Cpb (x:).
Remark 4 (Examples satisfying assumptions of Theorem 3p. Fix x. Let the reference be spherical with radial, strictly decreasing continuous density fu (u) o( with location m(x) and a positive definite scatter matrix Exx) whose whitened density uses the same radial generator as U, i.e.,
([ul))). Suppose Y X is elliptical
(v) x Φ
Then the map (1) = 2xx))112()) m(x)) and det [VyS = det 11/2. This setting includes the Gaussian case by taking dbr) x 22/2
To show that this setting indeed satisfies the assumptions of Theorem
we start from the transport
map Tn. By construction of the elliptical model, T2 (u) m(xx + E(x) 222 pushes Fy to Fyix== Moreover, Tn is the gradient of the convex quadratic
(w) fuu E(x) m(x)
so by Brenier-Knott-SSmin theory for the quadratic cost, Tr is the optimal transport from Fy to FylxEr-
Denote the inverse map by Sz(y) = 1(x)-1/2(() -m(x). Its Jacobian is constant, Sz) = 1/2) and therefore
=: Cu. > Theorem 3 assumes the existence of function je: 00) + (0, oc) such that
det (V,,S,())) dot
det det(V (v, ju (110 (v. x)))
25

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In the elliptical case Qvix = Sz(y) and det det(V,Ss(t)) Czs simply set ja(r)
Ce = dct((((2)11/)) for all 2 O.
Finally, since is strictly decreasing and ja(r) = C2 > is constant, the product r 6(r))=(r)
(x((r) is strictly decreasing in as required.
To show that Sp is indeed the optimal transport map, note that S. the gradient of convex quadratic function. Thus, it satisfies the Brenier optimal transport conditions for the Euclidean quadratic cost and. by Knott-Smith optimality criterion, is the vector quantile function (Knott & Smith 1984).
Conformal HDP Sets using OT Parameterization. While the CQR-like construction in Section robust and simple, its prediction sets are images of Euclidean spheres and thus topologically connected since, under Assumption and Assumption 2 Qrix
y is continuous by Carlierr et al. (2016 Corollary 2.1). This can be inefficient if for some EX, the true conditional distribution I
multimodal, for example a Gaussian mixture. To solve this problem. is possible to construct prediction sets using the level sets of estimated conditional density, which can naturally form disconnected regions.
This approach utilizes the change-of-variables formula and leveraging Qrix to recover the plug-in conditional density estimator
ply x) (O((Q)) det [V Xy)
This estimator then be used to define conformity scores. For each point (Y,, in the cal- ibration set Deal we calculate the score 9; P(Y. Xi). The prediction set for a new point Xtest the superlevel set of this estimated density, where the level calibrated ensure cover-
age. If $(1) I ... < ((n) are the ordered scores from the calibration set, we set the threshold T [[[m++)a])- Then, the HtD-style prediction region is given by:
Chpp (x) fy V: p))
By standard arguments, this set fulfills the marginal coverage guarantee P(y,x) (Y chod( 2 a. Crucially, if the learned map y recovers the true rank map, then p( x) recovers the true conditional density, and the resulting prediction set is exactly the true HPD region.
Related density-based approaches. The idea of using density estimation to construct conformal sets has been exploited in recent related works. For example, in the setting with y C R. CD-splir partition X' into multiple splits, leverage conditional density estimator fly x). and perform conformal calibration in split-wise manner to improve conditional coverage (lzbicki et al.) 2022). Furthermore, also in the setting with y R, SPICE learns neural conditional density deep splines and uses negative log-densityHHPD scores to construct the conformal sets (Diamant et al. 2024).
Remark 5. To construct conformal sets using density estimation, the estimator of ply x)) requires the Jacobian of Qyix- Even if Qy approximates Qyix. v may not necessary approx- imate well V, Vy Empirically, small errors in the Jacobian can be magnified in det(-), which distorts HPD superlevel sets. As shown in Section 6.1 in our experiments, Q approximated well the true quantile function. Nonetheless, we found the HDP approach of producing conformal sets empirically suboptimal w.r.t. the volume of the produced set and conditional coverage.
G IMPORTANCE OF CONVEX POTENTIAL
The fact that the transformation is cyclically monotone is crucial for defining a statistically mean- ingful notion of multivariate rank. Cyclical monotonicity is the extension of monotonicity in the multidimensional setting. The definition of multivariate ranks is discussed in the pioneering work
Hallin et al, (202T) and in studies by Galichon (2018); Carlier et al. (2016) 2017| 2022): ; Cher- nozhukov et al. 2017
26

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G.1 CYCLICAL MONOTONICITY
A subset S _ R x R is said to be cyclically monotone if, for any finite collection of points (()..)), (Xx, yb)} S S, the following inequality holds:
(1. X2 x + (y2. X2 X2) + +... (Yx,XI - S.
Nn) C R x IR is cyclically monotone if and only if the in- equality above holds for n. Equivalently, maximizes the empirical correlation -i= (x,, Vi) among all pairings of (x1, . xn) and [yI,.... ........,. J. equivalently, minimizes the empirical dis- tance Di=1 ||y: x.Ila. In other words, a finite subset is cyclically monotone and only the pairs Ys) form the solution the optimal assignment problem with cost lly; - x:|1?. Rockafel- lar's Lemma (1966) (Rockafellar 2015) establishes relation between cyclical monotonicity and convex functions.
A finite subset = (x1. : y1)
Theorem (Rockafellar (015)), Theorems 1). The subdifferential du of a convex function i on IRd enjoys cyclical monotonicity. Conversely, any cyclically monotone set S of Rd x B. is contained in the subdifferential Öh of some convex function vp on R
Theorem 6 (Brenier (1911). Let and v be two probability measures on Rd, with absolutely continuous with respect to the Lebesgue measure. Let T: Rd Rd be measurable map that pushes forward μ onto i.e.
T=u= =.
Then, there exists unique (up to p-almost everywhere equality) cyclically monotone map Vip : Rd + Ré, where is corvex function, such that T o with S R2² + Ie measure- preserving map, that is Syp= = μ. Moreover. Vip is the optimal transport map pushing μ forward to v for the quadratic cost c(z, y) = | - yll?, and it minimizes Iga [[T - xl? du(x) among all measurable maps satisfying T μ1
Brenier's theorem states that any measurable transport map T can be factorized as:
measure-preservvng rearrangement S, which redistributes mass within μ without altering it. monotone
A myooooallymoonotnne map Vy, which moves μ optimally (in the quadratic sense) to
When defining multivariate quantiles, we learn the cyclically monotone pushforward between the data distribution over Ed and the standard normal Gaussian distribution denoted μ (we could take the uniform distribution on the disk or the multivariate normal with mean and identity covariance). We assume that is absolutely continuous w.r.t. the Lebesgue measure.
Assume that we stake MVN(0.II, we obtain the (1 - a)) coverage sets, we take balls of radius Pai where Pa is the 1 a quantile of x² distribution with degrees of freedom and compute their pre-images under this map.
If transform which is not eyclically monotone, the regions that will obtain using this method will of course have the correct coverage, but might have wild shapes, because the measure preserving transform (appearing in Brenier's polar factorization) can be arbitrarily complex. Indeed, there infinitely many pushforward map between μ and v, but only the cyclically monotone pushfor- ward allows to define quantile function and the associated ranks and signs that are meaningful.
G.2 EXAMPLE OF NON CYCLICALLY- MAP
We illustrate this phenomenon in dimension d 2.
Assume that the data distribution is multivariate normal with non-singular covariance matrix T. We map it the multivariate distribution with density TLe5 (w) MVN 0, I), where Leb the Lebesgue measure.
The cyclically monotone transform V = 1-122u where 1122 is the Hermitian root-square of L (and (u) = (1/2)z² T 1/2 is a convex function). The Brenier polar factorization theorem shows that any pushforward -> μl can be decomposed as T 4h where is measure- preserving map for N(0, I). This measure preserving maps is not cyclically monotone and therefore might affect the ranks used in the construction.
27

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There many ways construct construction R². Let (a(t), u()) a two dimensional random process which evolves according to the the ODE:
preserving maps of MVN(00, I). We provide such
a(1) = vt. (x((), in = (v((),
where 24 (U P is the velocity field. Denote by Pt the pdf of (a(t), y(t) at time The continuity equation shows that
V.P. + (pewe) =0
The density is invariant under the flow of the ODE is V (pew)) 0 for all 0, in which = for all 2 O. Denote W/ Pt A canonical way to ensure V wy = is to set
wt, O, (z, y) and Way 0g (x,, y), where plays the role of Hamiltonian. If we start
from po the pdf of NNVNOO, the flow of the ODE
1 (z(1), ,(1)),, ((() 5(() --(1- (o(), ((())) Po
((1)
Po
is measure preserving for Po. for any 0. In other words, if we define (x+(t, Fo-+)(((0), v(0) the flow of the solutions of the ODE, Fo-s #Po = po- Along the flow of the solutions,
d ((x(t), 02 (a((), (()) ((t) (a)(((())
dt
so the Hamiltonian v is constant along the orbit: V (xx(t), (()) C. We illustrate this with the mixing
(xx v) = (x² + 11 - -2
(A., B.). we show the level set obtained by applying the eyclically monotone transform. In (...D.) show the level sets obtained using cyclically monotone transformation T, obtained by integrating the Hamiltonian flow until tile 1.5. (E., F.) show empirical density estimate from 10 points, sampled from different operators.
Results are presented in Figure 3
is clear that a transformation that is not not cyclically monotone does not define a notion of ranks sim- ilar to the described in Hallin et al. (2021). and cannot be used to construct conformalization procedure similar to ours.
H DETAILED EXPERIMENTAL RESULTS
H.1 OPTIMAL TRANSPORT METRICS
Wasserstein distances. We compute Wasserstein-2 and Sliced Wasserstein distances using the POT library Flamary et al. (202T).
KDE-L1. To estimate the L' distance between kernel density estimators, we draw 1000 samples from both Qyix and its approximation 1 We then fit Gaussian kernel density estimates to each sample and report the average pointwise L' difference between the two densities, evaluated at points drawn from Qrix
KDE-KL. The Kullback-Leible divergence is computed following the same procedure as KDE-L1. We report the average pointwise KI. divergence between the fitted densities at points drawn from Q
L2-UV. To compute the unexplained variance ratio, we sample n. points from Weest Fu and ng points from Fistt ~ Fx The L2-UV distance then defined
11 (Uheess Tiest) - QU (Mheett Frea))12
Σ
The This Mees
my Urea Quix (Ueeat Fress) QUix (Mrest Frest
H.2 OPTIMAL TRANSPORT EXPERIMENTS DATASETS
Banana Dataset. This dataset is largely used in vector quantile estimation for testing the non- lincarity of estimators, It was introduced in (Feldman et al. 2023) and used in (Carlier et al. 2017] Rosenberg et al. 2023). It represents banana-shaped random variable in R22 changing its position
28

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B.
24
12
025
-15
-075
-15
-1.0
C.
D.
F.
2.5
2.5
E
14
33
0.00
63
63
-17
-37
-100
-20
-20
-10
Figure 3: Two pushforward operators. (A., B.) Level sets latent distribution chosen be standard Gaussian. (B.) Pushforward of level sets by cyclically monotone operator. (D.) Pushforward of level sets by non cyclically operator, generated by flow. (E., F.) Empirical Density Estimates, based on points sampled from cyclically monotone operator (E.) and non cyclically monotone operator (F.).
29

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and skewness based on latent random variable from R². Data generative process can be described as:
Z ~~ X X~U[08,3221,
8 1~~ 2-0
--110_01,011
3
3 ~ 100,111
B |131|2
Y. 3/ (- 1-cos( ?)) + 1) rin((() sin(X),
Z
Yi 3X +rcos(pp)
Yo
X = x, =
Yi
We take X and as observed random variables.
Rotating Star. This dataset is inspired by (Rosenberg et al. (003)) rotating star example. Observed random variable represents three point star in I that rotates based on latent variable from R. Data generative process can be described as:
2
(un, u1) T)O,T), X X~
3.
arctan
11 aag
s(0) 1 3 cos()
cos
sin y
R(
sin
up
cos 4
Y
(0)mi) x- X,
where yo is rotation angle. We take X, Y as observed variables.
Glasses. This dataset is introduced in (Brando et 2022). represents modal distribution,
where random variable is in R. With X ~ 2[00 1], data generative process can be described as:
22 = (1 + 3X), 21 3
Beta(a 0.5, 3 1),
Y1 = 5 sin 21) + 2.5 + Yg = 5sin(=2) + 2.5 e,
7 Categorical 1), Y = (1- + 7Y2.
We take X, Y as observed variables.
Neal's funnel distribution.
The classical funnel distribution (Nal 2003) is defined on Rd+1 as J~~ N(0,00)
It l expxex)
I....d,
so that the joint density of (u, (u..,...., Ed) is
2
2(u,z) V2TO²
exp
20
V2Te"
exp
2
a
For large negative values of u. the conditional variance of the x;'s shrinks, yielding a narrow region (the "neck" of the funnel), whereas large positive produces very diffuse 2;'s 's (the "mouth"). This strong nonlinearity makes the distribution challenging for MCMC methods.
Multidimensional funnel. A natural generalization introduces k-dimensional scale vector & (Ui...., Ub) with
Uj N(O,o²),
Zze 1 ~ exp(:))), { 1=
that each by controls block of m Gaussian variables. The joint distribution then lives in di- mension kc(1 + m) and exhibits multiple funnel directions simultaneously. This aigh-dimessionl geometry frequently used stress test for MCMC and normalizing flow methods.
30

---

Banana, X=2.0
Star, X=1.047
Glasses, X=1.047
Funnel, X=1.047
Ground Iruth C-NQRy
Figure 4: Example of points sampled from reference distribution of all the datasets we study and points sampled from approximation constructed by C-NQRy method.
Dimension of funnel
B
12
14
16
E-NOR C-NOR, .+ C-NOR,
CA-NQR,
CA-NOR,
CFF
FNAVOR
VOR
Wasserstein alstanze
0.8
0.0
0.8
Figure 5: Log values of S-W2 on Neal's funnel distribution. We scale the dimension of funnel from 2 to 16.
H.3 EXTENDED RESULTS ON OPTIMAL TRANSPORT
Here we give extended results on our internal experiments of proposed quantile regression esti- mate.
Generative Model In Figure we plot examples of samples from different datasets. Dark blue points are sampled from underlying ground truth generative model. Green points represent samples from C-NQRy generative model.
Scalability To evaluate scalability, Figure reports the S-W2 metric on Neal's Funnel as the target dimension increases from 2 to 16. We observe that methods leveraging explicit c-transform com- putation (NQRy, NQRy, AC-NQRu, AC-NQRy) scale robustly with dimensionality and maintain high generative accuracy, while entropically relaxed variants (EC-NQR) fail to scale effectively. Furthermore, the proposed framework consistently achieves superior S-W2 metric compared to prior approaches (Rosenberg et al. 2023 Carlier et al. 2016), demonstrating its performance in high-dimensional quantile estimation.
Full set of metrics Lastly present full set of metrics for studied datasets:
Metrics for Glasses dataset is accessible Figure 11
Metrics for convex potential, that was trained on Glasses dataset can be found at Figure 12
Metrics for Star dataset is accessible at Figure g
Metries for convex potential, that was trained on Star dataset can be found Figure 10
Full set of metrics for Banana dataset is accessible at fig. 7 Metrics for convex potential, that was trained on Banana dataset can be found at fig.
H.4 DETAILED RESULTS OF THE CONFORMAL PREDICTION EXPERIMENTS
We present more detailed results on conditional coverage on real datasets, involving more variations of our methods and more nominal levels
31

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Methods. We include the HPD variant of method well as models estimating either the forward (U) or the inverse (Y) quantile map.
For methods labeled with Y, we model the function & with a neural network and have Qy (y, x) = Vybby,, x).. For methods labeled with U we model function yp and get Qr (y. x)) = V Vug x)..
Method Quantile corresponds using the Monge-Kantorovich rank construct the predictive gions, assuming that we have found exactly the mapping to the reference standard multidimensional normal distribution. In this particular case, the squared ranks follow the Chi-square distribution and the corresponding radius for the construction of the pullback-type prediction set can be found exactly.
The methods labeled with RF correspond to fitting our model to the residuals of - f(x) of base Random Forest predictor f. Base predictor uses 25% of the training data and remainder used train our model.
Implementation details. For baseline methods we use the original authors implementation, where available and their suggested values for hyperparameters. For our methods, we select the number of parameters for neural networks to be roughly 10% of the number of training samples. We tune the other hyperparameters for each dataset using separate data split and utilize the mean coverage error of the pullback sets at different levels of a performance measure. All experiments were replicated using random splits of the data into training, calibration, and test parts.
Discussion. The Quantile method fails achieve the nominal levels of conditional coverage, which suggests that a supporting measure like conformal prediction is indeed required. Unfortu- nately, HPD approaches do perform well many occasions, proving that density estimation in multiple dimensions is still difficult to solve problem.
Using base model and fitting quantile regression to the residuals instead of directly Y provides less variable results, but does not always improve performance of our methods.
32

---

OJa1the
190
24
f
sgens.
spenw.
6.9
6.4
bis,
bib.
6.4
blog. e= 0.1
blog. 0.2
blog, 0-3
0.3
Figure 6: Worst slab coverage at different nominal miscoverage levels for conformal prediction methods, achieved on large datasets.
33

---

Wassenstenn Fs
Siked Wasserstein Py
KDE KL. Fr
KDF KL 11
CAOA
20.12-
2.000
CMOR,
0.30
0.030
0.00-
0.050
0.00
0.00
0.003
0.020
R..0-
0.000
0.15
0.330
0.523
Mocel's
Wasserstain
Wasserstein
KL
0.200
0.10
0.6
0.150-
10
000
0.00
0.02
0.05
0.01
Medel's
Wasserstetn
Sliced Wasserstein
KDE
0.4-
CAOR,
EAGA
CFF HRVOH
CAOH,
CANON,
0.6
0.20
02-
02-
0.15
0.65
Notel' Idde
Veiee Indox
Vosdd indox
Modkts Inace
Wassertein Fuz
Sloed Wasversiein Fug
KDE Kuy
KDE L1 Fay
0.20
0.200-
0.175
0.38
0.030
0.250
6.6-
0.125-
0.34
0.32
0.155
Mocefs
Vedel Eidae
Wasserstain Frc
Fros
BACk
0.15
CAORL
2 0,300
0.20
CACA,
CANOT
VOl
6.15-
0.03
0.500
0.00
0.05
0.0
0.01
Wasserstein
Sliced Wassersteen
KDE KL
KDE L1 Fig.
5.
0.175
0.5
64325
0.150-
0.00
0
01
Mocéfa iidex
Vocil Fidex
Modela Fidax
Figure 7: Full set of metrics for Banana dataset.
34

---

Convex
Wasserstein Fy
Sliced Wasserseenn
KDE 1L. Pv
KDe KL. L. FL
0.85
6.14-
10.30
6:01
0.
0.025
0.
0.220
0,035
6:22
0,232
tniee
Wasserstein
KLFyy
2179
5.150
0.4
0.00
0.300
0.075
62-
3,02
6.00
0.5
Wasserstein
Sliced Wassersteth
FeaR
CA-109
0.30
C-MOA
CM
0.5
0.25
H9 Von
OA-RO
0.5
6.20-
6.15-
0.5
Mise's
Medif's I1ace
Made reex
Wussersiein Faze
Sliced Waiversiein Fu
KL
KDE L1 Fas
+
0.375
6.30
2ted Nusurnrr
0.150
0.22
0:33
6:15-
0.155
Vodel's bices
hoex
Frce
Wasserstein For
Foust
CA-
0.5
0.05
0.25
20
C-00%
von..
Sieee wonerster
0.2
0.4
0.2
0.04
0.2
62-
0.05
6.1
0.60
0.5
Wasseestwin
Slined Wasserstein
KDE KL Fuuu)
KDE L3 Fu.s
03
S5o0 Ma0000T
0.18
04
0:28
0.112
0.te
0.36
62.
0.00
0.204
Mode's ride
Medila Fizes
Medid Ireee
Mode roex
Figure 8: Full set of metrics for Banana dataset.
35

---

Wassentein Ps
Siieed Wasserstain Py
KDE KL. Fr
KDF KL 11
CAOA
2.200
CMOR,
0.25
0:3-
0.15
0.30
00.00
0.05
Nocel's
Wasserstein
Wasserstein
KLFFy
0.08
0.67
0.100
0.020
20.0
0.125
C 0.:15
0.00
0.08
Wasserstetn
Sliced Wasserstein
KDE
0.40-
0.45
CAOR,
EAGA
CFF
0.20-
0.020
CANH,
0.25
0.24
0.018
0.3
0.616
0.5-
18-
0.1
Motels index
Musel ReSEX
Vosee indox
Wasserstein Fur
Slieed Wassersiwin Fuge
KOE Fuy
KDE Fuy
0.100
0.13
0.138
0.21
0.030
0.66-
0.33
6.67
Mocets
Wasserstain Frce
Frost
0.25
0.25
CAORL
BACK
CA-OR,
0.510
6.30-
CAOR,
0.112
yor
0.5
0.25
6.15
0.4
0.000
0.30
0.3
0.008
Wasserstein
Steed Vasseesteen
KDE KL
KDE Fige
0.32
0.00
6.07
n.000
515
0:00
0.1
0.007
0.88.
0.00
012
0.008
607.
0.00
0.005
0.c05
Nocita Rdee
Mocél Ridex
Modill Fidax
Figure 9: Full set of metrics for Star dataset.
36

---

Convex
Visserslein Ps
Siied Wasserstein Py
KDE KL. Fy
KDF XL LL
0.25
CAOR,
0.3
0.5-
2010
05
20.00
0022
Misdetc
Wasserstain Fu
Fg
6.30-
0.72
0.024
0.00
0.00
0.00
0.26
2,00
0.04
2,00
0:32
Wasserstem
Slceed Wasserstein
EAGA
CA-ON
0.40
0.000
0.35
CAOR,
CPF
6.6-
0.55
CAOH,
CANO1,
0.50
Fre
0.25-
0.22-
6.35
0.20
2.14
155-40
0.38-
Pedkl's index
Vesee EndE
Wosid indox
Musels ndox
Wasserstein Fure
Slieed Wosversiein Fug
KGE Kuy
KDE Fuy
10
6.310
0.110
0.014
0.325
0.115
0.00
6.320-
0.023
6.115-
2.03
0.315
Modele
Frr.s
From
CAcn,
CAORL
FANOR
6.00
0.-
5,020
0.20
CANOO
Von
6.55-
2.035
0.15-
0.20
0.30
0.
Wasserstein
Sliced Vasserstee Ps.c
KDE KL
KDE L1 Fu
0.00
0.28
2.0 5..00
0.24
8.66-
6:22
0.55
0.00
2,00
8.c4
5.07
Voddl Fidee
Viciel Fidex
Modill
Mesela nde
Figure 10: Full set of metrics for Convex Star dataset.
37

---

Glasses
Wasserdten
Sliced Wassensein
KBe KL
KDe XL
0-4
0:15
2,035
5.00-
0.10
2,055
2,004
0.05
-2.011
2.02
0.00
Modee's
Wasserstein
Wasserstein Fg
0.334
0.32
0.035
0.332
0,000
0.330
0.306
0.00
6:22
2.05
2,00
0.60
0,033
Wasserstein
Sliced Wasserstel:
KDD
0:20
ENOR
CANOY
250
CANAA
CFF
0.8
CA-MOA,
200
0.2-
0.2
0.35
0.5
Voeee's FRBER
Madidfs IYSCE
Modet's indox
Mosi % ndox
Wasserstein Fure
Sleeed Wassersiein Fuc
KDE RL Fay
KOE Fuy
0.345
0200
0.15
0.240
ardd anvoveenee
0:330
0.20
6ox-
2.00-
0.225
0.30
Index
Modeets Fece
Iides
Mode
Ficas
CANA
CANOS,
0.30
0.27-
0.15
CANR,
Vor
0.25
0.20
6.15
2.015
0.10
0.15
6.16-
2,000
0.25
6.10
6:55
2.05
0.00
0.05
6.35
Wassentain
Slied Wasserstein
KDE Kuu.
KDELL
2.0235
0.08
5.015
5.0230
8.07
the B.Fucr
2,00-
2.8225
8.06
0.05
3.0335
602
0.00
5003
Viciel's Fidex
"accida Indoe
Medalaa Index
Mode's rdoe
Figure 11: Full set of metrics for Glasses dataset.
38

---

Convex
Vissserlaiin
Sliced Wassentein
KBd KL. KV
KDe XL Fr
ENOR
6.5
0.022
0.25
3.015
2,05
2.035
0:65
3.002
0.00
Modee's
Wasserstein
2.030
0.230
0.30
2.025
6.300
5.00
2.000
0.00
0.204
0.09
6.05
Wasserstem
Sliced Wassersteln
KID
ENOR
CAND1,
1.5
C-NCR
CFF
0.25
Whaye AATT
CA-MOA,
PRAVDA
7.5
Fa
0.4
0.25
0.5
Wonee's inDex
Manedts
Modet's indox
Molic % ndox
Wasserstein Fur
Sleed Wussersiein Fur
KDE KL Fay
KDE Fuy
2.045
0.0075
0.230
6.36-
Rred mncoveeee
Wideedter
0.15
3.035
0.24
2.022
0.0030
0.33
0.2/7
2.022
Model's Forat
hides
Mode
Fir.s
CANOS
2.020
0.25
CAOR,
CANA
CK-MOA
VOn
RK-
5 5.30
6.25
K0E Fr.4
6.15
2
0.10
5.20
0.05
0.05
+
+
Wassemstai Par.o
Sloed Wasserstein Forzs
KDE Fuu
KDE LDELLE Fig
3.6135
5.07
0:25
2.030
3.6155
0.00
2.025
5,0375
0.60
2.35
Vicie'a Fidex.
"Macidta Index
Medalaa Index
Mode's rioe
Figure 12: Full set of metrics for Convex Glasses dataset.
39


Reviews:
[{"review_id": "tAgCSqz3V1", "reviewer": "Reviewer_6k36", "summary": "The general methodology is to rely on mutlivariate quantiles of Hallin et al. to define a multvariate score, converted into univariate score, from which conformal prediction tools can be applied.\n\nThe novelty is the nature of the optimal transport solver used to compute the center outward map. Instead of relying on entropic-regularized solvers (Sinkhorn) like Klein et al, authors rely on the gradient of a partial input convex neural network (PICNN). The training leverages the amortization trick of Amos et al to accelerate the fitting of the dual potential. \n\nThe method is benchmarked on a few multivariate regression datasets, like the ones introduced by Rosenberg et al. The average Sliced-2 wasserstein-2 distance is computed to benchmark the method, and other metrics like coverage or log-volume of prediction sets are compared against competitors works or baselines.", "questions": "### Fig. 2\n\nWhy is the Wasserstein distance reported as negative?\n\n### Pareto front\n\nYou write:\n\n> The re-ranking step of RPB and RPBS allows to achieve a slightly\nsharper conditional coverage, but the increase in prediction sets volume make it a questionable trade-\noff.\n\nI think this observation mandates a Pareto front, to better position the relative strengths of each method.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "### Motivation\n\nThe paper extend the work of Thurin et al; Klein et al using PICNN, a parameterized approach.\n\n###  Exposition\n\nThe paper makes a good job at introducing the necessary tools (center outward quantiles, neural dual solvers, conformal prediction).\n\n### PICNN training\n\nICNN (and PICNN) training can be challenging. Various variants are tested. The results of Figure 3 and 4, show strong results: this is my main motivation for leaning toward acceptance. Fig 1. confirms that this PICNN can fit complex shapes. Since the work is done in low dimensions (the Y remaining much smaller than the X space) PICNN don't suffer as much from the curse of dimensionality.", "weaknesses": "### Novelty on the multivariate conformal prediction part \n\nThis is similar to the approach of two concurrent works (Thurin et al, Klein et al) that this paper acknowledges: using multivariate quantiles of Hallin et al, fit the (empirical) center outward map (or an approximation of thereof), and converting these generalized (multivariate) quantiles into 1D score .  \n\nThe main novelty is reliance on PICNN/ICNN instead of cubic linear solvers (Thurin et al) or entropic solvers (Klein et al), using other tricks documented in W2 Monge map learning.  \n\n### Clarity\n\nI had some issues other than navigate the experimental part, that lacked clarity at times. e.g Table 1 is not directly related to conformal prediction, it's simply measuring the distance between $T_{\\sharp}P$ and $Q$ (ie it's a sanity check). Also some issues with Fig 2 (see below).\n\nSimilarly, it's not clear to me which message Fig 2. is trying to convey.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EE1jB7XvRC", "reviewer": "Reviewer_Vkyx", "summary": "This paper combines neural optimal transport with conditional vector quantile regression to improve multivariate prediction and uncertainty estimation. The authors use convex neural networks to model transport maps, suggest employing amortized optimization to improve efficiency, and apply entropic regularization for stability. They then use the resulting quantile maps to build conformal prediction sets that represent uncertainty in multiple dimensions. While the approach brings together several existing ideas, it is implemented carefully and evaluated on both synthetic and real datasets.", "questions": "1.\tHow sensitive are the results to violations of the radial Jacobian assumption in Theorem 3?\n2.\tHow does the proposed approach compare against modern conditional flow or diffusion models for conditional sampling and prediction sets?\n3.\tCan the proposed approach handle large output dimensions, e.g., $d_y > 20$?\n4.\tThis paper introduces multiple methods: pb, rpb and hdp. Is there a suggestion for which of them to use in practice?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.\tThis work introduces a new, novel method (NQR) that connects neural optimal transport with VQR.\n2.\tOptimality guarantee (Theorem 3).\n3.\tThe writing is generally clear and well-structured.\n4.\tThe experimental results show that the proposed approach yields better performance compared to the baselines.", "weaknesses": "1.\tThe paper’s novelty is moderate rather and not groundbreaking. Most of its core components, such as PICNN/ICNN (Makkuva et al., 2020; Amos et al., 2017), amortized optimization (Amos, 2023), entropic regularization (Genevay et al., 2016), and recent reranking conformal methods (Thurin et al., 2025; Klein et al., 2025), have already been explored in earlier work. The main contribution lies in how these ideas are combined and adapted for the conditional setting. \n2.\tThe assumptions of Theorem 3, which is the main theoretical result of this paper, are hard to verify in practice.\n3.\tThe conformal prediction experiments should include comparisons with VQR (Carlier et al., 2016, 2017, 2020), STDR (Feldman et al. 2023) and PCP (Wang et al. 2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QNcZSuOVV7", "reviewer": "Reviewer_XoMS", "summary": "This paper extends univariate conformal quantile regression (CQR) to the multivariate setting through the use of transport-based quantiles. To achieve this, the authors propose implementing optimal transport (OT) maps within vector quantile regression via Neural OT. The approach relies on input convex neural networks (ICNNs) to enforce the monotonicity of conditional quantiles or ranks. In addition, amortized optimization is employed to fasten the repeated computations inherent to conditional optimal transport.", "questions": "**PART I - Framing contributions in the literature** \n\nI.1 *l.49 “While well studied in the univariate case, multivariate extensions are less developed and often reduce to coordinate methods that ignore the geometry of joint distributions (Dheur et al.)”*: The paper by Dheur et al. actually discusses several possible ways to extend conformal prediction (CP) to multivariate regression. It therefore does not seem to be the most appropriate reference to support the claim that existing methods “reduce to coordinate approaches.” You might instead cite works that genuinely adopt coordinate-wise strategies.\nIn addition, the paragraph currently gives the impression that transport-based quantiles are the only viable direction for multivariate CP, which is not the case, as you yourself note later (lines 343–349) and in the extended related-work section. The introduction should be revised to explicitly mention the existence of alternative multivariate CP frameworks.\n\nI.2 *l.815:”Despite these advances, none of the above techniques exploits the full geometric structure of multivariate quantiles”*: It would help to clarify what is meant by “the full geometric structure.” One could argue that transport-based quantiles belong to the L-CP family of conformity scores introduced by Dheur et al. (2025). Indeed, your method also maps data to a latent space (albeit of the same dimension), similarly to Fang et al. (2025) or Feldman et al. (2023). These approaches appear complementary rather than mutually exclusive, so a more nuanced comparison would strengthen the framing.\nFor example, the monotonicity of the transport map in your model preserves the ordering of multivariate samples, this property distinguishes your method from the above works. It would be interesting to explain in which settings this specific feature provides a tangible advantage.\n\nI.3 *On Neural OT for multivariate quantiles*: The work of Vedula et al. (2023) on Continuous VQR appears closely related, as it also implements PICNNs for Neural OT. However, the similarities and distinctions between your approach and theirs are not sufficiently discussed. We recommend adding a short paragraph explicitly positioning your method with respect to prior studies that apply Neural OT to transport-based quantiles, possibly including targeted numerical comparisons if computational differences are significant.\nYou may also wish to cite recent references such as [3], which learns multivariate quantiles with PICNNs for time series, and [2], which uses amortized optimization for repeated OT computations. Clarifying these links would considerably improve how your contribution is situated within the broader literature.\n\n**Part II - Numerical comparisons**\n\nII1. Providing a comparison between different implementation variants is a valuable practice that can guide future practitioners. However, I have several questions regarding the results presented in Section 7.1.\nIn the tables, it is unclear what formatting (bold or underlined) denotes — could you clarify this in the captions or text? More importantly, what are the main takeaways from these experiments? You compare several Neural OT–based implementations of conformal quantile regression, but it is not evident which one should be preferred in practice.\nIn Table 1, for instance, are EC-NQR and CPF the best-performing methods overall? Is there a meaningful difference between learning the quantile map and the rank map, or do these approaches yield comparable results? Also, why is the AC-NQR variant used in the experiments for conformal prediction? A short Discussion paragraph after the results section would greatly help interpret these findings and highlight the practical implications of the comparisons.\n\nII2. Furthermore, some of the experimental comparisons seem to involve methods that are not directly comparable. For example, in Figure 4, you compare your CQR variants based on signed residuals (PBS and RPBS) with your main proposals (PB and RPB). Yet these two groups of methods do not rely on the same base model: PB and RPB employ a neural network, whereas other works use a random forest. It is therefore not surprising that PB and RPB yield lower volumes, as neural networks generally achieve higher predictive accuracy. A fairer comparison would restrict attention to PBS and RPBS versus ELL and OTCP(+), which share a similar setup.\n\nII3. In line with this point, it would also be natural to compare your method with Feldman et al. (2023), since their approach uses a neural network–based quantile regression as the base model — arguably a closer alternative to your main proposals (PB and RPB). Finally, since your re-ranking approach falls within the L-CP framework introduced by Dheur et al. (2025), a comparison with some of the methods considered in that reference would also be informative.\n\n**PART III - Clarification**\n\nIII1. The denomination “Conditional vector quantile regression” used in the abstract seems inappropriate. Conditional quantiles refer to the quantiles of a distribution conditional on covariates, while quantile regression designates the learning of such conditional quantiles from data. This latter notion corresponds to the focus of your work, together with a new algorithmic framework to achieve it. Therefore, the term vector quantile regression would be a more accurate and conventional choice.\n\nIII2. Theorem 1 (iii): you write a probability at the end of your sentence that I do not understand:  what is the meaning of $\\leq$ between two vectors? What is F_U(u), when F_U is defined as a probability measure? I assume that the point (iii) should be a sentence similar to the point (i): there exists a mapping that pushes the conditional distribution Y/X=x to the reference distribution F_U. Also, Assumption (2) is not customary: [1] defines the rank function in a semi-discrete setting, a fact that might appear in a remark. \n\n[1] Ghosal, Promit, and Bodhisattva Sen. \"Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing.\" The Annals of Statistics 50.2 (2022): 1012-1037\n\n[2] Amos, Brandon, et al. \"Meta optimal transport.\" Proceedings of the 40th International Conference on Machine Learning. 2023\n\n[3] Kan, Kelvin, et al. \"Multivariate quantile function forecaster.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2022.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This paper effectively integrates recent advances in computational optimal transport into the study of transport-based quantiles. It leverages Partially ICNNs (PICNNs) from Neural OT, amortized optimization for efficient computation of dual conjugates, and entropic regularization. These methodological choices are particularly relevant for enabling the use of transport-based quantiles in machine learning applications, where computational efficiency is crucial. The application of Neural OT to conformal quantile regression is both well-motivated and timely. I also appreciate the level of detail provided in the appendices regarding the different Neural OT implementations and the comprehensive presentation of results across all evaluation metrics.", "weaknesses": "My main concern is that the introduction may give a somewhat misleading impression of the existing literature. The incorporation of Neural OT into transport-based quantiles is not entirely new (see, for example, Continuous VQR, Vedula et al., 2023). Since this is claimed to be the first main contribution, a more explicit discussion of these related works would be valuable, in particular, to clarify the computational differences between the approach presented in this paper and prior methods. Although these papers are properly cited in the Appendix, the comparison should appear more prominently in the main text to help readers understand the methodological positioning.\n\nThe connection between transport-based quantiles and conformal prediction is also not entirely new, and the framing of the second contribution could be clarified. As currently written, it appears quite close to the contributions of recent works such as Thurin et al. (2025) and Klein et al. (2025). My understanding is that the approach presented in this paper generalizes conformal quantile regression by learning the conditional distribution Y∣X through transport-based quantiles, in the spirit of Feldman et al. (2023). In contrast, the perspective adopted in Thurin et al. (2025) and Klein et al. (2025) is to combine multivariate scores with multivariate quantiles, which represents a different line of research. This distinction is not clearly reflected in the introduction, although it does emerge in the main text and seems to motivate your re-ranking variants.\n\nIn addition, the experimental section would benefit from including numerical comparisons with related approaches, such as Feldman et al. (2023), to better position the method empirically.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "T4Jxw5iw9v", "reviewer": "Reviewer_Rw7A", "summary": "This work proposed a parametric approach to estimate conditional vector quantiles, where the push-forward is parameterized through the gradient of a (partially) convex neural network. An amortized strategy is adopted to reduce computation (by following the unconditional case) and an application to split conformal prediction is discussed. Experiments on small-scale synthetic dataset and regression benchmarks confirm the effectiveness of the proposed methods.", "questions": "My main concerns are on the significance and correctness: \n\n- most results appear to be rather straightforward adaptions of prior works (Sections 4 and 5). This is not to say there isn't anything new, but the writing makes it feel like these are rather incremental changes. \n\n- I was not able to verify the proof of Theorem 3 and I think its assumptions may be too strong. In particular, for the elliptical example the authors need to verify the displayed equation in Theorem 3. \n\nI was intrigued by the authors claim that \"Neural OT as we propose here have never been scaled\" (Line 321) and \"never from a unique joint sampling using the framework of Carlier et al. as proposed in our work\" (Line 340). A quick google search revealed the following work \"Conditional Generative Quantile Networks via Optimal Transport\" (https://openreview.net/forum?id=BBxeo2Vuvbq), which seemed to be based on Carlier et al. and already proposed a neural approach for conditional vector quantile estimation. This latter work did not discuss conformal prediction but perhaps is still very relevant. \n\nOther comments: \n\nLine 148: \"In the multivariate setting, monotonicity requires the map to be the gradient of a convex function.\" How is monotonicity defined in the multivariate setting? Why is it needed and why does it require the map to be the gradient of a convex function? \n\nFigure 3, right plot: the y-axis (log V)/d can be as low as -0.7 (even -5 for the second plot). For moderately large d this means the volume of the conformal set is exceedingly small. Is this expected? \n\nLine 467: the resulting response dimensions are 16, 4, 2, and 2. The potential based OT approach is known to be difficult to scale (due to the evaluation of the conjugate potential). The authors' experiments seem to confirm this observation. How high the dimension can be for the proposed approach to remain effective?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "- a neural approach to estimate vector quantiles \n\n- a more versatile approach for conformal prediction", "weaknesses": "- limited novelty: the majority of the contributions are a straightforward extension of prior works, with no significantly new ingredients\n\n- possible gaps in Theorem 3: the proof on Line 1244-1246 needs more explanation. The proof seems to rely on the assumption that HPD superlevel sets are \"optimal\"? In Remark 3, how is the displayed equation in Theorem 3 satisfied? Here the left-hand side does not depend on y while the right-hand side does. Is this assumption too strong? \n\n- writing can be improved. A lot of details are deferred to the appendix, including the description of the proposed methods (Algorithms 1, 2 and 3). At times, it is hard to appreciate what are the significant modifications on the authors' side.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)