Vector Quantile Regression (VQR) Overview

• Vector Quantile Regression is an optimal transport framework that models full conditional distributions of vector responses using latent quantile indices and convex gradients.
• It employs mean-independence constraints and cyclical monotonicity to ensure a global, geometrically consistent quantile mapping without quantile crossing.
• Recent advances integrate entropic regularization and scalable Sinkhorn-type algorithms, broadening VQR applications to manifold and time series data.

Searching arXiv for recent and foundational papers on Vector Quantile Regression to ground the article in published work. Vector quantile regression (VQR) is an optimal-transport-based framework for modeling the full conditional distribution of a vector-valued response $Y$ given covariates $X$. It generalizes scalar quantile regression by introducing a latent vector quantile index $U$, typically uniform on $[0,1]^d$ or drawn from another reference law $\mu$, and by requiring the regression map $u \mapsto Q(x,u)$ to be monotone in the multivariate Brenier sense, namely as the gradient of a convex potential. In the conditional formulation most commonly used in VQR, the latent index is constrained by mean independence, $E[X\mid U]=E[X]$, rather than full independence; this allows a global representation of conditional dependence while preserving the optimal transport geometry that underlies vector quantiles (Carlier et al., 2014, Carlier et al., 2016).

1. Definition and latent-rank structure

The basic unconditional object is the vector quantile function of a random vector $Y\in\mathbb{R}^d$. Under the Brenier–McCann conditions, there exists a convex potential $\varphi$ such that

$Y=\nabla \varphi(U),$

where $X$0 has a prescribed reference law, often $X$1. This is the multivariate analogue of the scalar nondecreasing quantile function, with monotonicity replaced by cyclical monotonicity of the gradient of a convex function (Carlier et al., 2014, Pegoraro et al., 2023).

For conditional problems, a conditional vector quantile function is a measurable family $X$2 such that $X$3. Under regularity, $X$4, where $X$5 is convex in $X$6, and there is a strong representation

$X$7

In the fully specified conditional transport formulation, $X$8 is independent of $X$9; in VQR proper, the independence restriction is relaxed to mean independence, $U$0, which is weaker than $U$1 and is central to identification in affine-in-$U$2 models (Carlier et al., 2014, Carlier et al., 2016).

This distinction is fundamental. VQR is not coordinatewise quantile regression applied separately to each component of $U$3. Its monotonicity is global and geometric: for fixed $U$4, the graph of $U$5 is cyclically monotone, and when $U$6,

$U$7

This is the multivariate analogue of “no quantile crossing,” but it is expressed through convex analysis rather than through an ordering on $U$8 (Carlier et al., 2016).

2. Optimal transport formulation and convex duality

The foundational VQR program seeks a latent $U$9 that is maximally correlated with $[0,1]^d$0 subject to a prescribed law and the mean-independence constraint. In the notation of Carlier, Chernozhukov, and Galichon, the primal problem is

$[0,1]^d$1

with $[0,1]^d$2 typically the uniform law on $[0,1]^d$3 after centering $[0,1]^d$4 (Carlier et al., 2016). In the linear specification, one writes $[0,1]^d$5, with $[0,1]^d$6 required to be the gradient of a convex function for each $[0,1]^d$7 (Carlier et al., 2014).

The dual formulation introduces potentials $[0,1]^d$8 and $[0,1]^d$9 and a slack function $\mu$0 satisfying

$\mu$1

with

$\mu$2

At optimum, complementary slackness gives

$\mu$3

The multiplier $\mu$4 is the dual object associated with mean independence, and its appearance is what differentiates VQR from standard unconstrained optimal transport (Carlier et al., 2016).

A parallel formulation, emphasized in the entropic literature, uses the quadratic cost

$\mu$5

and couplings $\mu$6 between $\mu$7 and $\mu$8 that satisfy

$\mu$9

The feasible set is

$u \mapsto Q(x,u)$0

where $u \mapsto Q(x,u)$1 is the joint law of $u \mapsto Q(x,u)$2. This quadratic-cost formulation preserves the Brenier structure and provides the Schrödinger-type equations used by later entropic and Sinkhorn-based methods (Kato et al., 23 Mar 2026).

3. Specification, misspecification, and the scalar reduction

Under correct specification, VQR assumes an affine-in-$u \mapsto Q(x,u)$3 structure with joint convexity in the latent index. In one formulation,

$u \mapsto Q(x,u)$4

where $u \mapsto Q(x,u)$5 is convex in $u \mapsto Q(x,u)$6 for $u \mapsto Q(x,u)$7-almost every $u \mapsto Q(x,u)$8, $u \mapsto Q(x,u)$9, and $E[X\mid U]=E[X]$0. Under this structure, the regression map $E[X\mid U]=E[X]$1 is the Brenier map between the reference law and the conditional law of $E[X\mid U]=E[X]$2 (Carlier et al., 2016).

A central theoretical result is that VQR remains meaningful beyond correct specification. Under regularity conditions on the support and density of the joint law of $E[X\mid U]=E[X]$3, the dual problem admits a solution, and the optimal representation becomes

$E[X\mid U]=E[X]$4

Here $E[X\mid U]=E[X]$5 is the convex envelope of $E[X\mid U]=E[X]$6. Thus, under misspecification, the fitted object is not generally a genuine conditional Brenier map; rather, it is a best convex approximation in the sense of convex-envelope subgradients. This directly addresses a common misconception: VQR does not require correct affine specification in order to exist or to produce an interpretable latent-rank representation (Carlier et al., 2016).

In the scalar case $E[X\mid U]=E[X]$7, VQR connects tightly to Koenker–Bassett quantile regression. Classical quantile regression solves, for each $E[X\mid U]=E[X]$8,

$E[X\mid U]=E[X]$9

while the global VQR program becomes

$Y\in\mathbb{R}^d$0

The 2016 analysis shows that, in the univariate case, VQR is equivalent to classical quantile regression with an additional monotonicity constraint across quantile indices. This resolves the “$Y\in\mathbb{R}^d$1-by-$Y\in\mathbb{R}^d$2” incoherence of scalar quantile regression by embedding the whole conditional quantile family into a single global latent-factor program (Carlier et al., 2016).

4. Entropic regularization and structural results

Entropic VQR regularizes the coupling by adding a Kullback–Leibler penalty relative to the product measure $Y\in\mathbb{R}^d$3. With $Y\in\mathbb{R}^d$4,

$Y\in\mathbb{R}^d$5

subject to the same mean-independence constraint

$Y\in\mathbb{R}^d$6

This is a Schrödinger-type relaxation of VQR and can be interpreted as an entropic projection problem (Kato et al., 11 Feb 2026).

The corresponding dual involves three potentials,

$Y\in\mathbb{R}^d$7

with objective

$Y\in\mathbb{R}^d$8

Relative to standard entropic OT, the new ingredient is the vector-valued potential $Y\in\mathbb{R}^d$9, which couples linearly with $\varphi$0 through $\varphi$1 and enforces mean independence. Under suitable coercivity and invertibility assumptions, strong duality holds, dual maximizers exist, and the unique primal optimizer has exponential-family density

$\varphi$2

The dual potentials are unique up to the affine shift

$\varphi$3

(Kato et al., 11 Feb 2026).

The optimality conditions form a Schrödinger-type system. The scalar potentials $\varphi$4 and $\varphi$5 satisfy log-normalization equations, while $\varphi$6 is characterized by the additional moment equation

$\varphi$7

This moment equation has no counterpart in standard entropic OT and is the main analytical complication of entropic VQR (Kato et al., 11 Feb 2026).

Recent work established two further structural results. First, when supports are compact, $\varphi$8, each coordinate of $\varphi$9, and $Y=\nabla \varphi(U),$0 are real analytic. Second, in the Gaussian case, entropic VQR admits a closed-form solution: if $Y=\nabla \varphi(U),$1 and $Y=\nabla \varphi(U),$2 is jointly Gaussian, then the optimal coupling $Y=\nabla \varphi(U),$3 is again Gaussian, and the entropic approximation satisfies

$Y=\nabla \varphi(U),$4

This yields an explicit approximation rate toward unregularized VQR (Kato et al., 11 Feb 2026).

5. Algorithms, Sinkhorn-type methods, and large-scale computation

The earliest entropic numerical work on VQR emphasized smooth dual optimization rather than classical Sinkhorn scaling. In the discrete setting, the smoothed dual is a convex unconstrained program in the dual variables, and mean independence is encoded through the $Y=\nabla \varphi(U),$5 terms. Because the KL projection onto the mean-independence constraints lacks a closed form, exact multiplicative Sinkhorn updates are unavailable; this motivated gradient descent with Nesterov acceleration in the 2021 numerical study (Carlier et al., 2021).

The 2026 Sinkhorn paper introduced two algorithms directly adapted to entropic VQR. The first is a classical Sinkhorn-type block coordinate ascent. It alternates an implicit $Y=\nabla \varphi(U),$6-update solving, for each $Y=\nabla \varphi(U),$7,

$Y=\nabla \varphi(U),$8

followed by log-sum-exp updates for $Y=\nabla \varphi(U),$9 and $X$00, and then normalization to fix the affine ambiguity. The second, described as new in the literature, replaces the implicit solve by a projected gradient step

$X$01

where $X$02 is the $X$03-projection onto the set

$X$04

The projection admits an explicit form involving a finite-dimensional convex program with Huber penalty and can be solved by iterative reweighting (Kato et al., 23 Mar 2026).

Under compact supports, $X$05, and invertible $X$06, both algorithms have rigorous linear convergence. For the classical scheme,

$X$07

with explicit $X$08, and the iterates converge in $X$09. The modified projected-gradient scheme satisfies an analogous linear rate under the step-size condition

$X$10

A key innovation of this analysis is the derivation of explicit quantitative bounds on the dual potentials and Sinkhorn iterates (Kato et al., 23 Mar 2026).

In discrete implementations with $X$11 reference points for $X$12 and $X$13 samples for $X$14, the modified algorithm has per-iteration cost $X$15, plus $X$16 for the projection subroutine. Practical guidance in the same work emphasizes log-domain stabilization, centering of $X$17, normalization of potentials at every iteration, and the usual entropic trade-off: larger $X$18 yields faster convergence but more bias, while smaller $X$19 produces sharper quantiles at the cost of slower convergence and stricter step-size requirements (Kato et al., 23 Mar 2026).

A separate line of work pursued large-scale approximate VQR through relaxed dual objectives and neural embeddings. The 2022 nonlinear VQR paper replaces $X$20 by a learned embedding $X$21, yielding

$X$22

and optimizes a smooth entropic dual with two-sided minibatching on quantile levels and data points. The reported implementation maintains a fixed memory footprint, scales to millions of samples and thousands of quantile levels, and provides a post hoc “vector monotone rearrangement” that solves a $X$23 OT problem on the estimated quantile grid to enforce co-monotonicity exactly (Rosenberg et al., 2022). This also clarifies a practical controversy: relaxed dual objectives can violate monotonicity for finite $X$24, so exact monotonicity may require either the exact dual geometry or a posteriori rearrangement (Rosenberg et al., 2022).

6. Extensions, applications, and limitations

VQR has been extended beyond Euclidean regression in several directions. On Riemannian manifolds, the Euclidean quadratic cost is replaced by the squared geodesic cost

$X$25

and convex potentials are replaced by $X$26-concave functions. In this setting, the manifold conditional vector quantile function is

$X$27

with $X$28 on the manifold $X$29. The framework supports quantile estimation, conditional confidence sets, and likelihood computation on spaces such as $X$30 and $X$31 (Pegoraro et al., 2023).

For time series, “Nonparametric Vector Quantile Autoregression” develops a center-outward version of the theory. The one-step-ahead quantile map of $X$32 given the past is defined as a conditional center-outward quantile mapping, estimated by combining a Nadaraya–Watson conditional law estimator with a discrete quadratic-cost OT problem. Under mixing, convex-support, and density assumptions, the estimated map is uniformly consistent on compact subsets of the unit ball away from the origin, and the resulting prediction regions have asymptotically correct conditional coverage (González-Sanz et al., 3 Oct 2025).

Applications reported across the literature include multiple Engel curve estimation in the original VQR paper, anthropometric data from ANSUR II, the Iris dataset, synthetic Gaussian benchmarks with known dual values, and manifold-valued examples involving climate or geological phenomena on spheres and protein dihedral angles on tori (Carlier et al., 2014, Carlier et al., 2021, Kato et al., 23 Mar 2026, Pegoraro et al., 2023). The common methodological theme is that VQR provides a full conditional distributional representation rather than a point prediction or a family of marginal quantiles.

Several limitations recur across these papers. Computational complexity grows rapidly with the dimension of the response, because convexity or cyclical monotonicity must be enforced globally and discrete grids scale as $X$33 in many implementations (Rosenberg et al., 2022). Under misspecification, identification can be partial, with subgradient- rather than gradient-based representations (Carlier et al., 2016). In manifold and time-series settings, regularity assumptions such as absolute continuity, avoidance of cut loci, convex support, and mixing or ergodicity are central to uniqueness and consistency (Pegoraro et al., 2023, González-Sanz et al., 3 Oct 2025). Entropic regularization improves numerical stability but introduces bias, so VQR inherits the now standard trade-off between exact transport structure and computational tractability (Carlier et al., 2021, Kato et al., 23 Mar 2026).

Taken together, these results define VQR as a family of OT-based conditional distribution models centered on a latent rank-like variable $X$34, a multivariate monotonicity notion given by convex gradients or $X$35-concavity, and a mean-independence relaxation that links multivariate quantiles to regression structure. The modern literature has expanded that core from linear Euclidean models to misspecified settings, entropic duality, Sinkhorn-type algorithms with linear convergence, manifold-valued responses, scalable nonlinear approximations, and nonparametric autoregressive forecasting (Carlier et al., 2016, Kato et al., 11 Feb 2026, Kato et al., 23 Mar 2026).