On the optimal mapping of distributions

Knott, M.; Smith, Cyril Stanley

doi:10.1007/bf00934745

Cited by 204 publications

(147 citation statements)

References 2 publications

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“…, (X n,n , X n )). A similar bound is obtained for the second term in (9). We deduce by Gronwall's lemma that We recall a result proved in Rachev and Rüschendorf [14] giving L 2 -rates of convergence of empirical measures in the Wasserstein metric.…”

Section: Proof Of Proposition 43supporting

confidence: 78%

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

Fontbona

Guérin

Méléard

2008

Probab. Theory Relat. Fields

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In this paper, we consider nonlinear diffusion processes driven by space-time white noises, which have an interpretation in terms of partial differential equations. For a specific choice of coefficients, they correspond to the Landau equation arising in kinetic theory. A particular feature is that the diffusion matrix of this process is a linear function the law of the process, and not a quadratic one, as in the McKeanVlasov model. The main goal of the paper is to construct an easily simulable diffusive interacting particle system, converging towards this nonlinear process and to obtain an explicit pathwise rate. This requires to find a significant coupling between finitely many Brownian motions and the infinite dimensional white noise process. The key idea will be to construct the right Brownian motions by pushing forward the white noise processes, through the Brenier map realizing the optimal transport between the law of the nonlinear process, and the empirical measure of independent copies of it. A striking problem then is to establish the joint measurability of this optimal transport map with respect to the space variable and the parameters (time and randomness) making the marginals vary. We shall prove a general measurability result for the mass transportation problem in terms of the support of the transfert plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates.Key words and phrases: Landau type interacting particle systems, nonlinear white noise driven SDE, pathwise coupling, measurability of optimal transport, predictable transport process. MSC: 60K35, 49Q20, 82C40, 82C80, 60G07.

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Section: Proof Of Proposition 43supporting

confidence: 78%

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

Fontbona

Guérin

Méléard

2008

Probab. Theory Relat. Fields

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“…If c satisfies a twist condition, meaning x ∈ X −→ c(x, y 1 ) − c(x, y 2 ) has no critical points for y 1 = y 2 ∈ Y , then we shall see that not only is the minimizing γ unique, but its mass concentrates entirely on the graph of a single map f 1 : X −→ Y (a numbered limb system with one limb), thus solving a form of the transportation problem posed earlier by Monge [61,81]. This was proved in comparable generality by Gangbo [52] and Levin [66] (see also Ma, Trudinger and Wang [72]), building on the more specific examples of strictly convex cost functions c(x, y) = h(x − y) in X = Y = R n analyzed by Caffarelli [19], Gangbo and McCann [53,54], Rüschendorf [89,90] and in case h(x) = |x| 2 by Abdellaoui and Heinich [1], Brenier [17,18], Cuesta-Albertos, Matrán, and Tuero-Díaz [31,32], Cullen and Purser [34,35,87], Knott and Smith [64,93], and Rüschendorf and Rachev [91]. Adding further restrictions beyond this twist hypothesis allowed Ma, Trudinger, Wang [72,100], and later Loeper [68], to develop a regularity theory for the map f 1 : X −→ Y , embracing Delanoë [36], Caffarelli [20,21] and Urbas' [102] results for the quadratic cost, Gangbo and McCann's for its restriction to convex surfaces [55], and Wang's for reflector antenna design [106], which involves the restriction of c(x, y) = − log |x − y| to the sphere [57,107].…”

Section: Uniqueness Of Optimal Transportationmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

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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 Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) 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 (Lattice Theory. Revised Edition, 1948): 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 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. 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.

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“…One can show that in this case one gets a unique minimizer given by the gradient of a convex function; see [3], [11], [20], and the references therein. Note that the Kantorovich-Wasserstein metric defines the distance between two mass densities, by seeking the "cheapest" way to transport the mass from one domain to another with respect to the metric (3).…”

Section: Monge-kantorovich Problemmentioning

confidence: 99%

“…More specifically, the updating equation is given by (18) If SSD is used as the comparison term, has the form of (19) and if MI is used as the comparison term, is given by (20) where stands for 2-D convolution, and is the derivative of with respect to its first variable. Once the algorithm converges, we have the final warping function .…”

Section: Inimizermentioning

confidence: 99%

See 1 more Smart Citation

An Image Morphing Technique Based on Optimal Mass Preserving Mapping

Zhu

Yang

Haker

et al. 2007

IEEE Trans. on Image Process.

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Abstract-Image morphing, or image interpolation in the time domain, deals with the metamorphosis of one image into another. In this paper, a new class of image morphing algorithms is proposed based on the theory of optimal mass transport. The 2 mass moving energy functional is modified by adding an intensity penalizing term, in order to reduce the undesired double exposure effect. It is an intensity-based approach and, thus, is parameter free. The optimal warping function is computed using an iterative gradient descent approach. This proposed morphing method is also extended to doubly connected domains using a harmonic parameterization technique, along with finite-element methods.

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On the optimal mapping of distributions

Cited by 204 publications

References 2 publications

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

Optimal transportation, topology and uniqueness

An Image Morphing Technique Based on Optimal Mass Preserving Mapping

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