Existence and uniqueness of monotone measure-preserving maps

By Robert J. McCann. 1995

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Citations in the corpus

Deep Imitative Models for Flexible Inference, Planning, and Control (2018, methodology): “R2P2’s use of pushforward distributions (McCann et al., 1995), employed in other invertible generative models (Rezende & Mohamed, 2015; Dinh et al., 2016) allows it to efﬁciently minimize both false positives and false negatives (type I and type II errors) (Neyman & Pearson, 1933).”
Some Applications of Mass Transport to Gaussian-Type Inequalities (2002, background): “The result, as improved by McCann [13], is as follow: If μ and ν are Borel probability measures on R and μ is absolutely continuous with respect to the Lebesgue measure, then there exists a convex function φ such that ∇φ pushes μ forward to ν.”
On the inverse implication of Brenier-Mccann theorems and the structure of (P 2 (M),W 2 ) (2011, methodology): “Theorem 1.8 (Cabré-McCann) Let ϕ : M → R ∪ {−∞} be a c-concave function and x, y ∈ M such that y ∈ ∂c+ϕ(x).”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “The result is also closely related to the theory of optimal transportation of measures (and can be deduced from the results of Brenier [16] and McCann [45]).”
A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones (2012, background): “mal transport theory, it is well known that there exists an a.e. unique map T: E!K(which we call the Brenier map) such that T= r˚where ˚is convex, T 2BV(Rn;K);and det(rT(x)) = 1 for a.e. x2E(see e.g. [5, 30, 29, 1]). Moreover, since T is the gradient of a convex function with positive Jacobian, rT(x”
Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media (2006, unknown): “Thus expρ : H1,2ρ −→ N is well-defined, and surjective as a consequence of the Brenier/McCann theorem [18, 39], which associates to each ρ ∈ P2(Rd) and ρ′ ∈ P2(Rd) a convex function ψ(x) + |x|2/2 on Rd (taking values in R ∪ {+∞}) whose gradient pushes ρ forward to ρ′ [39].”
Optimal transportation with capacity constraints (2012, unknown): “…From [M97], if s fails to be a diﬀeomorphism but is given by the gradient of a convex function, equation (18) continues to hold f -a.e. Recall [B91, M95] that for the cost (cid:10) c ( x, y ) = x · y , the optimal maps S n ( x ) and T n ( y ) have the form x (cid:11)→ ∇ ψ ( x ) and y (cid:11)→ ∇ φ…”
The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems (2004, background): “If F : Rd-R and G : Rd-R are two probability densities, it follows from Brenier [6] and McCann [16] that there exists a convex function j : Rd-R such that the measure GðyÞ dy is the image measure of FðxÞ dx under the gradient map rj: We say that the map T :¼ rj transports FðxÞ dx onto GðyÞ dy: The m”
Transport Inequalities for Log-concave Measures, Quantitative Forms, and Applications (2015, background): “Given two probability measures μ and ν on R with densities F and G, respectively, we know from Brenier [9] and McCann [21] that there exists a convex function ψ such that”
Vector Quantile Regression (2014, background): “Regarding the proof, the ﬁrst assertion of the theorem is a consequence of fundamental results due to McCann ([25]) (as, e.g, stated in [34], Theorem 2.32) who in turn reﬁned the fundamental results of [3].”
Best constant in Sobolev trace inequalities on the half-space (2006, methodology): “By a result of Brenier [5], further refined by McCann [21], there exists a function T = ∇ϕ : Rn+ → Rn+, with ϕ convex, such that T maps the measure F (y)dy onto G(y)dy, which means by definition that, for all measurable set B, ∫ B G(y)dy = ∫ T−1(B) F (y)dy, (12) or, equivalently, for all measurable ”
Bayesian inference with optimal maps (2011, background): “(After [33, 45]) Let μ0 and μ be Borel probability measures on R with μ0 vanishing on subsets of R having Hausdorff dimension less than or equal to n−1.”
Transport Inequalities. A Survey (2010, result): “For any ρ0, ρ1 which are compactly supported and such that Tc(ρ0, ρ1) ∞, we have (34) F (ρ1) − F (ρ0) ≥ κTc(ρ0, ρ1) + ∫ Rk (T (x) − x) · ∇[s′(ρ0) − s′(µ)](x) ρ0(x) dx where T is the Gangbo-McCann map which pushes forward ρ0 to ρ1.”
A mass transportation approach to quantitative isoperimetric inequalities (2010, background): “Postponing a rigorous discussion to the proof of Theorem 2.3, we recall that Brenier-McCann Theorem [8, 38 , 39] ensures the existence of a convex, continuous function ϕ : R n → R, whose gradient T =∇ ϕ pushes forward the probability density |E| −1 1E(x)dx into the probability density |K| −1 1K (y)”
Stability of entropic optimal transport and Schrödinger bridges (2021, background): “McCann showed in [36] that cyclical monotonicity singles out a particular transport map for quadratic cost c on Rd even if the optimal transport problem (here the 2-Wasserstein distance) is infinite, thus extending Brenier’s map to this setting; see also [52, pp.”
Regularity of Potential Functions of the Optimal Transportation Problem (2005, unknown): “For a more general result, based on the Aleksandrov argument [1], see [16, 21].”
Chapter 17 - Euclidean Structure in Finite Dimensional Normed Spaces (2001, background): “Existence and uniqueness of the Brenier map were proved in [26] (see also [125] for a different proof and important generalizations).”