A Parallel Method for Earth Mover’s Distance

Li, Wuchen; Ryu, Ernest K.; Osher, Stanley; Yin, Wotao; Gangbo, Wilfrid

doi:10.1007/s10915-017-0529-1

Cited by 67 publications

(92 citation statements)

References 16 publications

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“…Despite tremendous theoretical insight gained into the problem of optimally transporting one density to match another one, its numerical solution remains challenging, particularly when the densities live in spaces of dimension four or more. In small-dimensional cases, there are many state-of-the-art approaches that effectively compute the global solution; see, e.g., [30,55,67,46] and the recent survey [74]. Due to their reliance on Euclidean coordinates, those techniques require spatial discretization, which makes them prone to the curse of dimensionality.…”

Section: Optimal Transport and MLmentioning

confidence: 99%

A machine learning framework for solving high-dimensional mean field game and mean field control problems

Ruthotto

Osher

et al. 2020

Proc. Natl. Acad. Sci. U.S.A.

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148

109

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Mean field games (MFG) and mean field control (MFC) are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. Stateof-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse-of-dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing numerical methods.

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Section: Optimal Transport and MLmentioning

confidence: 99%

A machine learning framework for solving high-dimensional mean field game and mean field control problems

Ruthotto

Osher

et al. 2020

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

148

109

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show abstract

“…It is one of the prime motivations for this paper. We also note the minimizer satisfies the following form [17]:…”

Section: Unnormalized Optimal Transportmentioning

confidence: 99%

“…Experiment 4. Consider again the two dimensional problem, however this time we choose ρ 0 and ρ 1 to be the cats in [17]. Our results are summarized in Figure 4.…”

Section: Figures 2a -2cmentioning

confidence: 99%

Unnormalized optimal transport

Gangbo

Osher

et al. 2019

Journal of Computational Physics

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We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by . Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampére type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm [6]. This solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities. The L1 version of this metric was shown in [23] (which is a precursor of our work here) to have desirable properties.

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“…A key tool in our analysis of forward operator error is the Earth Mover's Distance. Below we summarize the presentation in [22].…”

Section: Earth Mover's Distancementioning

confidence: 99%

Diagnosing Forward Operator Error Using Optimal Transport

2019

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We investigate overdetermined linear inverse problems for which the forward operator may not be given accurately. We introduce a new tool called the structure, based on the Wasserstein distance, and propose the use of this to diagnose and remedy forward operator error. Computing the structure turns out to use an easy calculation for a Euclidean homogeneous degree one distance, the Earth Mover's Distance, based on recently developed algorithms. The structure is proven to distinguish between noise and signals in the residual and gives a plan to help recover the true direct operator

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A Parallel Method for Earth Mover’s Distance

Cited by 67 publications

References 16 publications

A machine learning framework for solving high-dimensional mean field game and mean field control problems

A machine learning framework for solving high-dimensional mean field game and mean field control problems

Unnormalized optimal transport

Diagnosing Forward Operator Error Using Optimal Transport

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