--- title: 'Monge-Kantorovich Ranks: Theory and Applications' url: https://www.emergentmind.com/topics/monge-kantorovich-ranks type: topic --- # Monge-Kantorovich Ranks: Theory and Applications 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 [1412.8434], [1806.01238]. ## 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 [1806.01238]. McCann’s theorem ensures that there exists a convex potential \(\psi\) on \(\mathbb{R}^d\) such that the gradient \[ Q^{\circ}(u) = \nabla\psi(u), \qquad u\in\overline{\mathbb{B}}_d, \] pushes \(\mathcal{U}(\overline{\mathbb{B}}_d)\) forward to \(P\), so \((Q^{\circ})_{\#}\mathcal{U}(\overline{\mathbb{B}}_d)=P\). Writing \(\phi=\psi^*\) for the Legendre transform of \(\psi\), \[ \phi(x)=\psi^*(x):=\sup_{u\in\overline{\mathbb{B}}_d}\{\langle u,x\rangle-\psi(u)\}, \] the reverse map is \[ F^{\circ}(x)=\nabla\phi(x), \qquad x\in\mathbb{R}^d, \] and it pushes \(P\) forward to the spherical uniform: \[ (F^{\circ})_{\#}P=\mathcal{U}(\overline{\mathbb{B}}_d). \] Moreover, \(\|F^{\circ}(x)\|\le 1\) for all \(x\), and under mild regularity the gradients are almost-everywhere inverses: \[ \nabla\psi(\nabla\phi(x))=x \quad P\text{-a.s.}, \qquad \nabla\phi(\nabla\psi(u))=u \quad \mathcal{U}\text{-a.s.} \] [1806.01238]. In the quadratic-cost Monge–Kantorovich problem, if \(P\) has finite second moment then \(F^{\circ}\) coincides \(P\)-a.s. with the \(L^2\)-optimal Brenier map, and its graph is cyclically monotone. Equivalently, \[ \sum_{i=1}^k \langle x_i, F^{\circ}(x_i)\rangle \ge \sum_{i=1}^k \langle x_i, F^{\circ}(x_{i+1})\rangle \] for any cycle \((x_1,\dots,x_k)\) with \(x_{k+1}=x_1\) [1806.01238]. ## 2. Ranks, signs, depth, and univariate reduction For an observation \(X\), the rank and sign are defined from the transport image \(F^{\circ}(X)\). The radial component \[ R=\|F^{\circ}(X)\|\in[0,1] \] is the center-outward analog of a univariate rank, interpreted as the probability content of the smallest center-outward quantile region containing \(X\). The angular component \[ S=\frac{F^{\circ}(X)}{\|F^{\circ}(X)\|}\in\mathbb{S}^{d-1}, \] with \(S=0\) if \(R=0\), is the multivariate sign. Under \(P\), \(R\sim \mathrm{Uniform}[0,1]\), \(S\) is uniform on \(\mathbb{S}^{d-1}\), and \(R\) and \(S\) are independent [1806.01238]. The earlier Monge–Kantorovich formulation expresses the same idea in terms of a reverse transport \(R_P(y)\) from the distribution \(P\) to a reference distribution \(F\). When \(F\) is the spherical uniform \(U_d\) on the unit ball, the MK rank of \(y\) is the vector \(R_P(y)\in S^d\), its scalar radial rank is \[ r_P(y):=\|R_P(y)\|, \] and the MK sign is \[ S_P(y):=\frac{R_P(y)}{\|R_P(y)\|}\quad\text{whenever }R_P(y)\neq 0. \] The corresponding center-outward ordering is \[ y_2 \geq^{\mathrm{MK}}_{P} y_1 \quad\Longleftrightarrow\quad \|R_P(y_2)\|\le \|R_P(y_1)\| \] [1412.8434]. A fundamental benchmark is the reduction to \(d=1\). In the center-outward formulation, \[ F^{\circ}(x)=2F(x)-1, \] so \[ R=|2F(X)-1| \] and the sign becomes \(\mathrm{sign}(X-\mathrm{median})\), recovering traditional ranks and signs [1806.01238]. In the MK depth formulation, the univariate rank map is \(R_P(x)=2P(x)-1\), and the resulting MK depth equals Tukey halfspace depth: \[ D^{\mathrm{MK}}_{P}(x)=\min(P(x),1-P(x)) \] [1412.8434]. The depth interpretation is central in the original formulation. For spherical reference \(U_d\), the MK \(\tau\)-quantile contour is \[ \mathcal{K}_P(\tau):=Q_P(\mathcal{S}(\tau)), \] the MK depth region is \[ \mathbb{K}_P(\tau):=Q_P(\mathbb{S}(\tau)), \] and the MK depth of \(y\) is Tukey halfspace depth evaluated at \(R_P(y)\). In \(d=1\), 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 [1412.8434]. ## 3. Empirical construction and computation For a sample of size \(n\), the empirical center-outward map is built on a regular grid in \(\overline{\mathbb{B}}_d\). One factors \[ n=n_R n_S + n_0, \] with \(n_R\) radii, \(n_S\) directions, and \(n_0\) copies of the origin. The grid consists of radii \[ r_j = \frac{j}{n_R+1}, \qquad j=1,\dots,n_R, \] directions \(u_\ell\), \(\ell=1,\dots,n_S\), forming an “as uniform as possible” set on \(\mathbb{S}^{d-1}\), and \(n_0\) copies of \(0\). The empirical target is the discrete uniform on this augmented shell-and-sphere grid [1806.01238]. Observed points \(X_i\) are assigned to grid points \(y_i\) by solving the quadratic-cost assignment problem \[ \min_T \sum_{i=1}^n \|X_i-T(X_i)\|^2, \] where the minimum runs over bijections from the sample to the grid, or equivalently over permutations \(\pi\) minimizing \[ \sum_{i=1}^n \|X_{\pi(i)}-y_i\|^2. \] The solution is cyclically monotone. The paper lists two computational routes: the Hungarian algorithm with complexity \(O(n^3)\), and auction algorithms with complexity \(O(n^2)\) with a parameter \(\varepsilon\), together with the instruction to verify optimality and, if needed, decrease \(\varepsilon\) [1806.01238]. The assignment defines the empirical map \(F_n^{\circ}(X_i)=y_i\), hence empirical ranks and signs \[ R_{i,n}=\|F_n^{\circ}(X_i)\|, \qquad S_{i,n}=\frac{F_n^{\circ}(X_i)}{\|F_n^{\circ}(X_i)\|}, \] with \(S_{i,n}=0\) if \(R_{i,n}=0\). If \(n_0>1\), ties occur at the origin; the paper notes that a small random jitter on a tiny inner sphere breaks ties and restores injectivity [1806.01238]. The computational study reported implementations handling \(n\) up to \(20{,}000\) in \(d=2\). After assignment, a smooth extension is computed via convex analysis and a projected gradient method; the smoothing parameter is obtained in \(O(n^3)\) via Karp’s minimum mean cycle algorithm [1806.01238]. 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 [1412.8434]. ## 4. Distribution-freeness, ancillarity, and semiparametric use A defining statistical property is finite-sample distribution-freeness. Under absolutely continuous \(P\), the vector \[ \big(F_n^{\circ}(X_1),\dots,F_n^{\circ}(X_n)\big) \] is uniformly distributed over all permutations of the grid points, with repeats for the \(n_0\) origins. Consequently, empirical ranks \(R_{i,n}\) and signs \(S_{i,n}\) are strictly distribution-free [1806.01238]. 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 \(F_n^{\circ}(X_i)\) is independent of it by Basu’s theorem. The \(\sigma\)-field generated by \[ F_n^{\circ}(X_1),\dots,F_n^{\circ}(X_n) \] 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 [1806.01238]. 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 \(P\), which yields procedures robust beyond ellipticity [1806.01238]. The earlier MK depth framework emphasizes a related point from the reference-measure side: if \(Y\sim P\), then the MK rank \(R_P(Y)\) is distributed according to the reference \(F\), and in the spherical case \(R_P(Y)\sim U_d\). That distribution-free pushforward property is the basis for rank and sign constructions adapted to general multivariate distributions rather than restricted spherical families [1412.8434]. ## 5. Smooth extension, Glivenko–Cantelli theory, and quantile geometry The empirical map \(F_n^{\circ}\) is initially defined only at observed points. At those points, a discrete Glivenko–Cantelli result holds: if \(X_1,\dots,X_n\) are i.i.d. with \(P\) in a broad class including convex supports and locally bounded densities, then \[ \max_{1\le i\le n}\|F_n^{\circ}(X_i)-F^{\circ}(X_i)\|\to 0 \quad\text{almost surely}. \] 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 [1806.01238]. To obtain a map on all of \(\mathbb{R}^d\), the paper constructs a convex piecewise-linear potential \[ \phi(x)=\max_{1\le j\le n}\{\langle x,y_j\rangle-\psi_j\}, \] with suitable weights \(\psi_j\) such that \(\nabla\phi(x_i)=y_i\) for all \(i\). Smoothness is then introduced via the Moreau–Yosida regularization \[ \phi_\varepsilon(x)=\inf_{z\in\mathbb{R}^d}\left\{\phi(z)+\frac{1}{2\varepsilon}\|z-x\|^2\right\}, \qquad \varepsilon>0, \] and one sets \(T_\varepsilon(x)=\nabla\phi_\varepsilon(x)\). For sufficiently small \(\varepsilon\), \(T_\varepsilon(x_i)=y_i\), cyclical monotonicity is preserved, and the range remains within the unit ball [1806.01238]. The resulting extension is Lipschitz with constant \(1/\varepsilon\). The largest admissible smoothing parameter is \[ \varepsilon_0=\frac{1}{2}\min_i\left\{\langle x_i,y_i\rangle-\psi_i-\max_{j\neq i}(\langle x_i,y_j\rangle-\psi_j)\right\}, \] which is computed via Karp’s minimum mean cycle algorithm in \(O(n^3)\). Choosing \(\varepsilon=\varepsilon_0\) yields the smoothest extension with Lipschitz constant \(1/\varepsilon_0\), and the bound is stated to be sharp within this construction; in \(d=1\), it attains the minimal possible Lipschitz constant [1806.01238]. This produces a continuous empirical map \(\bar F_n^{\circ}:\mathbb{R}^d\to\overline{\mathbb{B}}_d\) with uniform convergence \[ \sup_{x\in\mathbb{R}^d}\|\bar F_n^{\circ}(x)-F^{\circ}(x)\|\to 0 \quad\text{almost surely}. \] Population quantile regions and contours are defined by \[ \mathbb{C}(q)=Q^{\circ}(q\cdot\overline{\mathbb{B}}_d) =\{x:\|F^{\circ}(x)\|\le q\}, \] \[ \mathcal{C}(q)=Q^{\circ}(q\cdot\mathbb{S}^{d-1}) =\{x:\|F^{\circ}(x)\|=q\}. \] Under mild regularity, these regions are closed, connected, and strictly nested; contours are continuous hypersurfaces of Hausdorff dimension \(d-1\). The median set is \[ \mathbb{C}(0)=\bigcap_{0