Monge-Kantorovich Ranks: Theory and Applications

• Monge–Kantorovich ranks are transport-based statistical ranks that map multivariate data onto a spherical uniform reference, establishing a center-outward ordering.
• They decompose the transport image into a radial component (rank) and an angular component (sign), linking multivariate rank concepts to traditional univariate ranks.
• The framework supports distribution-free inference and robust nonparametric tests, with efficient computation using optimal assignment algorithms like the Hungarian method.

Monge–Kantorovich ranks are transport-based statistical ranks obtained by mapping a distribution of interest onto a fixed reference measure and then reading the image point in the reference space as a center-outward coordinate. In the multivariate setting, the leading construction uses the spherical uniform distribution on the closed unit ball as reference, so that the transported image decomposes into a radial component, interpreted as rank, and an angular component, interpreted as sign. This framework was introduced for multivariate depth, quantiles, ranks, and signs through optimal transport, and was later recast in a geometric center-outward form with stronger inferential guarantees and without moment assumptions (Chernozhukov et al., 2014, Barrio et al., 2018).

1. Transport-theoretic definition

Let $P$ be the distribution of $X\in\mathbb{R}^d$, absolutely continuous with respect to Lebesgue measure. The center-outward reference is the spherical uniform distribution on the closed unit ball, denoted $\mathcal{U}(\overline{\mathbb{B}}_d)$, where $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$. This reference is the product of a uniform radius $R$ on $[0,1]$ and a uniform direction $S$ on the unit sphere $\mathbb{S}^{d-1}$, independent of each other (Barrio et al., 2018).

McCann’s theorem ensures that there exists a convex potential $\psi$ on $\mathbb{R}^d$ such that the gradient

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

pushes $X\in\mathbb{R}^d$1 forward to $X\in\mathbb{R}^d$2, so $X\in\mathbb{R}^d$3. Writing $X\in\mathbb{R}^d$4 for the Legendre transform of $X\in\mathbb{R}^d$5,

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

the reverse map is

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

and it pushes $X\in\mathbb{R}^d$8 forward to the spherical uniform: $X\in\mathbb{R}^d$9 Moreover, $\mathcal{U}(\overline{\mathbb{B}}_d)$0 for all $\mathcal{U}(\overline{\mathbb{B}}_d)$1, and under mild regularity the gradients are almost-everywhere inverses: $\mathcal{U}(\overline{\mathbb{B}}_d)$2 (Barrio et al., 2018).

In the quadratic-cost Monge–Kantorovich problem, if $\mathcal{U}(\overline{\mathbb{B}}_d)$3 has finite second moment then $\mathcal{U}(\overline{\mathbb{B}}_d)$4 coincides $\mathcal{U}(\overline{\mathbb{B}}_d)$5-a.s. with the $\mathcal{U}(\overline{\mathbb{B}}_d)$6-optimal Brenier map, and its graph is cyclically monotone. Equivalently,

$\mathcal{U}(\overline{\mathbb{B}}_d)$7

for any cycle $\mathcal{U}(\overline{\mathbb{B}}_d)$8 with $\mathcal{U}(\overline{\mathbb{B}}_d)$9 (Barrio et al., 2018).

2. Ranks, signs, depth, and univariate reduction

For an observation $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$0, the rank and sign are defined from the transport image $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$1. The radial component

$$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$2

is the center-outward analog of a univariate rank, interpreted as the probability content of the smallest center-outward quantile region containing $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$3. The angular component

$$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$4

with $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$5 if $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$6, is the multivariate sign. Under $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$7, $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$8, $$\overline{\mathbb{B}}_d:= \{u\in\mathbb{R}^d:\|u\|\le 1\}$$9 is uniform on $R$0, and $R$1 and $R$2 are independent (Barrio et al., 2018).

The earlier Monge–Kantorovich formulation expresses the same idea in terms of a reverse transport $R$3 from the distribution $R$4 to a reference distribution $R$5. When $R$6 is the spherical uniform $R$7 on the unit ball, the MK rank of $R$8 is the vector $R$9, its scalar radial rank is

$[0,1]$0

and the MK sign is

$[0,1]$1

The corresponding center-outward ordering is

$[0,1]$2

(Chernozhukov et al., 2014).

A fundamental benchmark is the reduction to $[0,1]$3. In the center-outward formulation,

$[0,1]$4

so

$[0,1]$5

and the sign becomes $[0,1]$6, recovering traditional ranks and signs (Barrio et al., 2018). In the MK depth formulation, the univariate rank map is $[0,1]$7, and the resulting MK depth equals Tukey halfspace depth: $[0,1]$8 (Chernozhukov et al., 2014).

The depth interpretation is central in the original formulation. For spherical reference $[0,1]$9, the MK $S$0-quantile contour is

$S$1

the MK depth region is

$S$2

and the MK depth of $S$3 is Tukey halfspace depth evaluated at $S$4. In $S$5, and for spherical or elliptical families after affine standardization, MK depth coincides with halfspace depth; for more general distributions, MK contours can account for non convex features of the distribution of interest (Chernozhukov et al., 2014).

3. Empirical construction and computation

For a sample of size $S$6, the empirical center-outward map is built on a regular grid in $S$7. One factors

$S$8

with $S$9 radii, $\mathbb{S}^{d-1}$0 directions, and $\mathbb{S}^{d-1}$1 copies of the origin. The grid consists of radii

$\mathbb{S}^{d-1}$2

directions $\mathbb{S}^{d-1}$3, $\mathbb{S}^{d-1}$4, forming an “as uniform as possible” set on $\mathbb{S}^{d-1}$5, and $\mathbb{S}^{d-1}$6 copies of $\mathbb{S}^{d-1}$7. The empirical target is the discrete uniform on this augmented shell-and-sphere grid (Barrio et al., 2018).

Observed points $\mathbb{S}^{d-1}$8 are assigned to grid points $\mathbb{S}^{d-1}$9 by solving the quadratic-cost assignment problem

$\psi$0

where the minimum runs over bijections from the sample to the grid, or equivalently over permutations $\psi$1 minimizing

$\psi$2

The solution is cyclically monotone. The paper lists two computational routes: the Hungarian algorithm with complexity $\psi$3, and auction algorithms with complexity $\psi$4 with a parameter $\psi$5, together with the instruction to verify optimality and, if needed, decrease $\psi$6 (Barrio et al., 2018).

The assignment defines the empirical map $\psi$7, hence empirical ranks and signs

$\psi$8

with $\psi$9 if $\mathbb{R}^d$0. If $\mathbb{R}^d$1, ties occur at the origin; the paper notes that a small random jitter on a tiny inner sphere breaks ties and restores injectivity (Barrio et al., 2018).

The computational study reported implementations handling $\mathbb{R}^d$2 up to $\mathbb{R}^d$3 in $\mathbb{R}^d$4. After assignment, a smooth extension is computed via convex analysis and a projected gradient method; the smoothing parameter is obtained in $\mathbb{R}^d$5 via Karp’s minimum mean cycle algorithm (Barrio et al., 2018). In the earlier MK formulation, the discrete–discrete case likewise reduces to optimal assignment, while smooth–smooth settings are connected to convex-potential solvers such as Benamou–Brenier’s fluid-mechanics algorithm (Chernozhukov et al., 2014).

4. Distribution-freeness, ancillarity, and semiparametric use

A defining statistical property is finite-sample distribution-freeness. Under absolutely continuous $\mathbb{R}^d$6, the vector

$\mathbb{R}^d$7

is uniformly distributed over all permutations of the grid points, with repeats for the $\mathbb{R}^d$8 origins. Consequently, empirical ranks $\mathbb{R}^d$9 and signs $X\in\mathbb{R}^d$00 are strictly distribution-free (Barrio et al., 2018).

The same construction yields an ancillarity statement. The order statistic, understood as the sample viewed as an unordered set, is minimal sufficient and complete for the nonparametric model, while $X\in\mathbb{R}^d$01 is independent of it by Basu’s theorem. The $X\in\mathbb{R}^d$02-field generated by

$X\in\mathbb{R}^d$03

is described as essentially maximal ancillary: it carries all information about parameters of interest orthogonal to the nuisance density. In the paper, this is the finite-sample analog of semiparametric efficiency preservation, and invariance under data-driven order-preserving transformations is identified as the multivariate counterpart of univariate monotone-invariance in the sense of Hallin–Werker, although the full multivariate invariance theory is left for further development (Barrio et al., 2018).

These properties motivate rank-based procedures. The paper explicitly states that center-outward ranks and signs enable the construction of valid distribution-free tests, including multi-sample, regression, and independence procedures, and support semiparametric procedures that preserve efficiency. In elliptical models, center-outward ranks coincide with Mahalanobis ranks; their consistency extends to general absolutely continuous $X\in\mathbb{R}^d$04, which yields procedures robust beyond ellipticity (Barrio et al., 2018).

The earlier MK depth framework emphasizes a related point from the reference-measure side: if $X\in\mathbb{R}^d$05, then the MK rank $X\in\mathbb{R}^d$06 is distributed according to the reference $X\in\mathbb{R}^d$07, and in the spherical case $X\in\mathbb{R}^d$08. That distribution-free pushforward property is the basis for rank and sign constructions adapted to general multivariate distributions rather than restricted spherical families (Chernozhukov et al., 2014).

5. Smooth extension, Glivenko–Cantelli theory, and quantile geometry

The empirical map $X\in\mathbb{R}^d$09 is initially defined only at observed points. At those points, a discrete Glivenko–Cantelli result holds: if $X\in\mathbb{R}^d$10 are i.i.d. with $X\in\mathbb{R}^d$11 in a broad class including convex supports and locally bounded densities, then

$X\in\mathbb{R}^d$12

As a consequence, empirical ranks and signs consistently estimate their population counterparts, and empirical quantile contours consistently reconstruct the population contours at the observed points (Barrio et al., 2018).

To obtain a map on all of $X\in\mathbb{R}^d$13, the paper constructs a convex piecewise-linear potential

$X\in\mathbb{R}^d$14

with suitable weights $X\in\mathbb{R}^d$15 such that $X\in\mathbb{R}^d$16 for all $X\in\mathbb{R}^d$17. Smoothness is then introduced via the Moreau–Yosida regularization

$X\in\mathbb{R}^d$18

and one sets $X\in\mathbb{R}^d$19. For sufficiently small $X\in\mathbb{R}^d$20, $X\in\mathbb{R}^d$21, cyclical monotonicity is preserved, and the range remains within the unit ball (Barrio et al., 2018).

The resulting extension is Lipschitz with constant $X\in\mathbb{R}^d$22. The largest admissible smoothing parameter is

$X\in\mathbb{R}^d$23

which is computed via Karp’s minimum mean cycle algorithm in $X\in\mathbb{R}^d$24. Choosing $X\in\mathbb{R}^d$25 yields the smoothest extension with Lipschitz constant $X\in\mathbb{R}^d$26, and the bound is stated to be sharp within this construction; in $X\in\mathbb{R}^d$27, it attains the minimal possible Lipschitz constant (Barrio et al., 2018).

This produces a continuous empirical map $X\in\mathbb{R}^d$28 with uniform convergence

$X\in\mathbb{R}^d$29

Population quantile regions and contours are defined by

$X\in\mathbb{R}^d$30

$X\in\mathbb{R}^d$31

Under mild regularity, these regions are closed, connected, and strictly nested; contours are continuous hypersurfaces of Hausdorff dimension $X\in\mathbb{R}^d$32. The median set is

$X\in\mathbb{R}^d$33

a compact convex set of Lebesgue measure zero, and a point for $X\in\mathbb{R}^d$34 under the stated regularity (Barrio et al., 2018).

6. Scope of the term and later extensions

The expression “Monge–Kantorovich ranks” is used in several related but non-identical senses. In multivariate statistics, it denotes the transport-based ranks and signs associated with the map from $X\in\mathbb{R}^d$35 to a reference measure on the unit ball, together with the associated depth and quantile structures (Chernozhukov et al., 2014, Barrio et al., 2018). The 2018 center-outward treatment explicitly contrasts its geometric construction with the Monge–Kantorovich approach of Chernozhukov et al. (2017): the geometric center-outward approach adopts McCann’s geometric existence and a spherical uniform target, avoids moment restrictions, and builds empirical maps with smooth cyclically monotone extensions (Barrio et al., 2018).

A distinct use appears in the optimal-transport duality literature on infinite-dimensional spaces. In “On the Role of Cylindrical Functions in Kantorovich duality,” the term “Monge-Kantorovich ranks” is not used explicitly. There, “rank” refers instead to the finite rank of linear projections $X\in\mathbb{R}^d$36 and $X\in\mathbb{R}^d$37 defining cylindrical functions in the Kantorovich dual problem; increasing rank yields a consistent finite-dimensional approximation scheme for infinite-dimensional optimal transport (Zaal, 2015). A potential source of confusion is therefore terminological rather than conceptual: the “rank” in that setting is the dimension of a projection, not a statistical rank assigned to observations.

Another extension arises in one-dimensional differentiable sorting. “Differentiable Ranks and Sorting using Optimal Transport” formulates sorting as an OT problem on $X\in\mathbb{R}^d$38 between an input measure and a presorted target grid. The optimal coupling defines generalized K-ranks, K-CDFs, K-sorts, and K-quantiles, and entropic regularization yields differentiable S-ranks and S-sorts via Sinkhorn scaling. In that work, the construction is explicitly one-dimensional and algorithmic, aimed at differentiable learning pipelines rather than multivariate center-outward inference (Cuturi et al., 2019).

A further extension concerns image data. “Monge-Kantorovich quantiles and ranks for image data” first embeds images into a tangent space using linear optimal transport, then applies MK quantiles and ranks in a Log-PCA latent space with spherical-uniform reference on the unit ball. The paper defines an MK rank function $X\in\mathbb{R}^d$39, inner and outer depths for images, and uses the framework for descriptive analysis, outlier detection, and statistical testing (Thurin, 4 Mar 2025). This suggests that the center-outward OT paradigm is not confined to $X\in\mathbb{R}^d$40 point clouds, but can be transferred to structured objects once a suitable transport-linearized representation is available.