Optimal Transport Map Research Articles

From the heat measure to the pinned Wiener measure on loop groups

For each L p -Wasserstein distance ( p > 1 ) with the cost function induced by the L 2 -distance on loop groups, we show that there exists a unique optimal transport map solving the Monge–Kantorovich problem when the initial probability measure is absolutely continuous with respect to the heat kernel measure. In particular, this provides us a family of measurable maps on loop groups which push the heat kernel measure forward to the pinned Wiener measure.

Dual theory of choice with multivariate risks

We propose a multivariate extension of Yaari's dual theory of choice under risk. We show that a decision maker with a preference relation on multidimensional prospects that preserves first order stochastic dominance and satisfies comonotonic independence behaves as if evaluating prospects using a weighted sum of quantiles. Both the notions of quantiles and of comonotonicity are extended to the multivariate framework using optimal transportation maps. Finally, risk averse decision makers are characterized within this framework and their local utility functions are derived. Applications to the measurement of multi-attribute inequality are also discussed.

The Monge problem in Rd

We first consider the Monge problem in a convex bounded subset of Rd. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of Rd.

Optimal transport with convex obstacle

We consider the Monge transportation problem when the cost is the squared geodesic distance around a convex obstacle. We show that there exists at least one—and in general infinitely many—optimal transport maps.

A Remark on the Potentials of Optimal Transport Maps

Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.

On Hölder continuity-in-time of the optimal transport map towards measures along a curve

AbstractWe discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space ($(\mathscr{P}_p(\mathbb{R}^d),W_p)$)). We prove that in most cases such a map is no more than 1/p-Hölder continuous.

On displacement interpolation of measures involved in Brenier’s theorem

We prove that in the Wasserstein space built over R the subset of measures that does not charge the non-differentiability set of convex functions is not displacement convex. This completes the study of Gigli on the geometric structure of measures meeting the sharp hypothesis of the refined version of Brenier’s Theorem. Optimal transport is nowadays a central tool in many fields of analysis, differential geometry and probability theory (see e.g. the books by Villani [11, 12]). It takes its origin in the problem of Monge, asking for the shortest way to displace an amount of soil from one place of the Euclidean space to a heap of soil at another place. This problem induces a very natural distance between probability measures – the Wasserstein distance – where the measures represent the heaps of soil. Wasserstein distance is somewhat complementary with the Lebesgue L norms because it corresponds to an “horizontal” displacement: in first approximation, for two localized probability measures on R, the Wasserstein distance is the distance between their barycenters. Brenier’s Theorem [4] on monotone rearrangement of maps of R has become the very core of the theory of optimal transport. It gives a representation of the optimal transport map in term of gradient of convex functions. A very enlightening heuristic on (P2(R ),W2) is proposed in [7] where it appears with an infinite differential structure and the Wasserstein distance is seen as a Riemannian-like distance. This point of view has raised a lot of developments, among which the gradient flow theory presented in [2]. In [6], Gigli explores the Riemannian-like structure and proposes to think of the measures meeting the hypothesis of a refined version of Brenier’s Theorem as the regular points of the Wasserstein space. In this paper we show that the set of those transport-regular measures is not geodesically convex (Theorems 2.2 and 2.5). It is quite surprising because it is well-known that the subset made of absolutely continuous measures (the most notorious transport-regular measures) is geodesically convex. It answers a question suggested by Gigli [6, Remark 2.12]. The paper is built as follows: we first recall some main results on the quadratic Monge(-Kantorovich) problem. The key idea of our result can be found in Lemma 1.6 while Proposition 1.8 gives a way, together with Proposition 1.7, to characterize transport-regular measures. In the second (and last) part, we prove the main theorem (Theorem 2.2) and its generalization (Theorem 2.5) thanks to explicit constructions. 2010 Mathematics Subject Classification. Primary 28A75.

Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds

In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results.

PROJECTIONS ONTO THE CONE OF OPTIMAL TRANSPORT MAPS AND COMPRESSIBLE FLUID FLOWS

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentropic Euler equations. The key ingredient is a characterization of the polar cone to the cone of optimal transport maps.

Mass Transportation with LQ Cost Functions

We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier's Theorem proving existence and uniqueness of an optimal transport map. In the controllable case, we show that the optimal transport map has to be the gradient of a convex function up to a linear change of coordinates. We give regularity results and also investigate the non-controllable case.

The Monge problem for strictly convex norms in $\mathbb{R}^d$

We prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of ℝ_d_ under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures.

Counterexamples to Continuity of Optimal Transport Maps on Positively Curved Riemannian Manifolds

Counterexamples to continuity of optimal transport maps on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, and Wang [16] is not guaranteed by positivity of sectional curvature.

Mass Transportation on Sub-Riemannian Manifolds

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier–McCann’s theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows us to write a weak form of the Monge–Ampère equation.

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

AbstractGiven a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so‐called Ma‐Trudinger‐Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C4‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc.

Measurability of optimal transportation and strong coupling of martingale measures

We consider the optimal mass transportation problem in $\mathbb{R}^d$ with measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.

The Optimal Partial Transport Problem

Given two densities f and g, we consider the problem of transporting a fraction \({m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}\) of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by |x − y|2, we will see that uniqueness of solutions holds for \({m \in [\|f\wedge g\|_{L^1},\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}\) . This extends the result of Caffarelli and McCann in Ann Math (in print), where the authors consider two densities with disjoint supports. The free boundaries of the active regions are shown to be (n − 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets \({{\Omega,\Lambda \subset \mathbb {R}^n}}\) , and bounded away from zero and infinity on their respective supports, \({C^{0,\alpha}_{\rm loc}}\) regularity of the optimal transport map and local C1 regularity of the free boundaries away from \({{\Omega\cap \Lambda}}\) are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions.

On the regularity of solutions of optimal transportation problems

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.

Variational Heuristics for Optimal Transportation Maps on Compact Manifolds

Variational derivation of the expression of the solution of Monge’s problem posed on compact manifolds (possibly with boundary), assuming all data are smooth, the solution is a diffeomorphism and the cost function satisfies a generating type condition.

Regularity of optimal transport on compact, locally nearly spherical, manifolds

Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M, we look for a smooth optimal transportation map G, pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge–Ampère type satisfied by the potential function u, such that G = exp(grad u). This approach boils down to proving an a priori upper bound on the Hessian of u, which was done on the flat torus by the first author. The recent local C2 estimate of Ma–Trudinger–Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image G(m) of a generic point m ∈ M, uniformly away from the cut-locus of m; he checked a fourth-order inequality satisfied by the squared distance cost function, proving its so-called (strict) regularity. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in C2 norm—specifying and proving a conjecture stated by Trudinger.

Absolute continuity of Wasserstein geodesics in the Heisenberg group

In this paper we answer to a question raised by Ambrosio and Rigot [L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261–301] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the end-points is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolute continuity of Wasserstein geodesic also holds for Alexandrov spaces with curvature bounded from below.