arXiv subjects

Robert J. McCann

Publications and source records attributed to Robert J. McCann.

49 records · Page 3Linked to original sources

arXiv2010

Optimal transportation, topology and uniqueness

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff [14]: give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γoutside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph / antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

math.PR↗

arXiv2010

Hölder continuity for optimal multivalued mappings

Gangbo and McCann showed that optimal transportation between hypersurfaces generally leads to multivalued optimal maps - bivalent when the target surface is strictly convex. In this paper we quantify Hölder continuity of the bivalent map optimizing average distance squared between arbitrary measures supported on Euclidean spheres.

math.AP↗

arXiv2010

Regularity of optimal transport maps on multiple products of spheres

This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved [KM2]. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps. This together with the result of Liu, Trudinger and Wang [LTW] also implies higher regularity (C^{1,α}/C^\infty) of optimal maps for more smooth (C^α/C^\infty)) densities. These are the first global regularity results which we are aware of concerning optimal maps on non-flat Riemannian manifolds which possess some vanishing sectional curvatures. Moreover, such product manifolds have potential relevance in statistics (see [S]) and in statistical mechanics (where the state of a system consisting of many spins is classically modeled by a point in the phase space obtained by taking many products of spheres). For the proof we apply and extend the method developed in [FKM1], where we showed injectivity and continuity of optimal maps on domains in R^n for smooth non-negatively cross-curved cost. The major obstacle in the present paper is to deal with the non-trivial cut-locus and the presence of flat directions.

math.AP↗

arXiv2010

The Ma-Trudinger-Wang curvature for natural mechanical actions

The Ma-Trudinger-Wang curvature --- or cross-curvature --- is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be expressed in terms of the associated Jacobi fields. Using this expression, we show the least action corresponding to a harmonic oscillator has zero cross-curvature, and in particular satisfies the necessary and sufficient condition \Athreew\ for the continuity of optimal maps. We go on to study gentle perturbations of the free action by a potential, and deduce conditions on the potential which guarantee either that the corresponding cost satisfies the more restrictive condition \Athrees\ of Ma, Trudinger and Wang, or in some cases has positive cross-curvature. In particular, the quartic potential of the anharmonic oscillator satisfies \Athrees\ in the perturbative regime.

math.AP↗

arXiv2010

Extremal doubly stochastic measures and optimal transportation

This article connects the theory of extremal doubly stochastic measures to the geometry and topology of optimal transportation. We begin by reviewing an old question (# 111) of Birkhoff in probability and statistics [4], which is to give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Following work of Douglas, Lindenstrauss, and Beneš and Štěpán, Hestir and Williams [15] found a necessary condition which is nearly sufficient; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph / antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. Surprisingly, this characterization can be used to resolve the uniqueness question for optimal transportation on manifolds with the topology of the sphere.

math.PR↗

arXiv2010

Pseudo-Riemannian geometry calibrates optimal transportation

Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like currents in spaces with indefinite metrics.

math.DG↗

arXiv2010

Rectifiability of Optimal Transportation Plans

The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

math.AP↗

arXiv2009

When is multidimensional screening a convex program?

A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can choose only pure strategies, we identify a structural condition on the preference b(x,y) of agent type x for product type y -- and on the principal's costs c(y) -- which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal's problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal's optimum strategy -- and similarly of the strategy maximizing the expected welfare of the agents when the principal's profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang's necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.

math.OC↗

arXiv2009

Continuity and injectivity of optimal maps for non-negatively cross-curved costs

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target density f^-(y) is bounded away from zero and infinity on its support clV \subset R^n, which is strongly c-convex with respect to U', and the transportation cost c is non-negatively cross-curved, we deduce continuity and injectivity of the optimal map inside U' (so that the associated potential u belongs to C^1(U')). This result provides a crucial step in the low/interior regularity setting: in a subsequent paper [15], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. The present paper also provides an argument required by Figalli and Loeper to conclude in two dimensions continuity of optimal maps under the weaker (in fact, necessary) hypothesis A3w [17]. In higher dimensions, if the densities f^\pm are Hölder continuous, our result permits continuous differentiability of the map inside U' (in fact, C^{2,α}_{loc} regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang [33].

math.AP↗

arXiv2009

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on $R^d$ are studied. These equations constitute gradient flows for the perturbed information functionals $F[u] = 1/(2α) \int | D (u^α) |^2 dx + λ/2 \int |x|^2 u dx$ with respect to the $L^2$-Wasserstein metric. The value of $α$ ranges from $α=1/2$, corresponding to a simplified quantum drift diffusion model, to $α=1$, corresponding to a thin film type equation.

math.AP↗

arXiv2008

Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)

The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativity is preserved for products of two manifolds; (2) both A3w and cross-curvature nonnegativity are inherited by Riemannian submersions; and (3) the $n$-dimensional round sphere satisfies cross-curvature nonnegativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature nonnegativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for complex projective space CP^n.

math.DG↗

arXiv2007

Continuity, curvature, and the general covariance of optimal transportation

Let M and \bar M be n-dimensional manifolds equipped with suitable Borel probability measures ρand \barρ. Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \in C^4(M \times \bar M) to guarantee smoothness of the optimal map pushing ρforward to \barρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M \times \bar M. Hölder continuity of optimal maps was established for rougher mass distributions by Loeper, still relying on a key result of Trudinger & Wang which required certain structure on the domains and the cost. We go on to develop this theory for mass distributions on differentiable manifolds -- recovering Loeper's Riemannian examples such as the round sphere as particular cases -- give a direct proof of the key result mentioned above, and revise Loeper's Hölder continuity argument to make it logically independent of all earlier works, while extending it to less restricted geometries and cost functions even for subdomains M and \bar M of R^n. We also give new examples of geometries satisfying the hypotheses -- obtained using submersions and tensor products -- and some connections to spacelike Lagrangian submanifolds in symplectic geometry.

math.DG↗

arXiv2007

On the cost-subdifferentials of cost-convex functions

We are interested in the cost-convex potentials in optimal mass transport theory, and we show by direct and geometric arguments the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions, under the assumptions A0, A1, A2, and A3W on cost functions introduced by Ma, Trudinger, and Wang. The connectivity of contact sets of optimal transport maps follows as a direct corollary. Our approach is quite different from the previous result of Loeper which he obtained as the first step toward his Hoelder regularity theory of potential functions, and which was based upon approximation using the regularity theory of Ma, Trudinger, and Wang. The result in this paper improves his result, by relaxing certain geometrical assumptions on the domains and targets; it also completes his Hoelder regularity theory of potential functions on the round sphere, by making it self-contained.

math.AP↗