Numerical solution of the Optimal Transportation problem using the Monge–Ampère equation

Benamou, Jean-David; Froese, Brittany D.; Oberman, Adam M.

doi:10.1016/j.jcp.2013.12.015

Cited by 192 publications

(217 citation statements)

References 41 publications

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“…Our implementation is based on Chartrand, Wohlberg, Vixie, and Bollt (2009) and ideas from Benamou, Froese, and Oberman (2014) and uses the known fact that the Brenier map is the only mapping (i) that will transform the one given density into another given density and (ii) that can be written as the gradient of a convex function. The constraint that the original density f (y) be mapped to Φ (x) by the map x = T (y) can be expressed using the usual change of variables formula:…”

Section: Methodsmentioning

confidence: 99%

“…This characterization of the Brenier map actually even relaxes any requirement of y having a finite variance. Numerous numerical methods to find T (y) are available in the literature (e.g., Villani (2003), Villani (2009), Benamou and Brenier (2000), Chartrand, Wohlberg, Vixie, andBollt (2009), Benamou, Froese, andOberman (2014)). We show that this optimal transportation problem is directly related to the determination of nonlinear independent components that best represent the data.…”

Section: Outlinementioning

confidence: 99%

See 1 more Smart Citation

A nonlinear principal component decomposition

Gunsilius¹,

Schennach²

2017

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The idea of summarizing the information contained in a large number of variables by a small number of "factors" or "principal components" has been widely adopted in economics and statistics. This paper introduces a generalization of the widely used principal component analysis (PCA) to nonlinear settings, thus providing a new tool for dimension reduction and exploratory data analysis or representation. The distinguishing features of the method include (i) the ability to always deliver truly independent factors (as opposed to the merely uncorrelated factors of PCA); (ii) the reliance on the theory of optimal transport and Brenier maps to obtain a robust and efficient computational algorithm and (iii) the use of a new multivariate additive entropy decomposition to determine the principal nonlinear components that capture most of the information content of the data.

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Section: Methodsmentioning

confidence: 99%

Section: Outlinementioning

confidence: 99%

A nonlinear principal component decomposition

Gunsilius¹,

Schennach²

2017

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show abstract

“…Alternatively, there are continuous solvers, based on the polar factorization theorem and the Monge-Ampère equation (e.g. [20,14,7]). These need not handle the full product space, but work directly with a transport map and thus can solve large problems more efficiently.…”

Section: Background and Motivationmentioning

confidence: 99%

“…Assuming full support is also common for continuous solvers (e.g. [20,7]). This can be ensured by adding a small constant measure.…”

Section: Remark 62 (Truncation) Note That the Algorithms Lemon-ns Amentioning

confidence: 99%

A Sparse Multiscale Algorithm for Dense Optimal Transport

Schmitzer

2016

J Math Imaging Vis

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Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport, we provide a framework to verify global optimality of a discrete transport plan locally. This allows the construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multi-scale scheme. Any existing discrete solver can be used as internal black-box. We explicitly describe how to select the sparse sub-problems for several cost functions, including the noisy squared Euclidean distance. A significant reduction of run-time and memory requirements is observed.

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“…We also describe how to construct first-order algorithms to compute critical points to the corresponding optimization problems and how to apply the proposed methods to image interpolation. Notice that due to this generalized form, the associated Euler-Lagrange equations are not of Monge-Ampère type and second-order algorithms based on this formulation [30], and recently [19,6,5,7], can not be used in our context. The dynamical optimal transportation formulation is also the only one that can handle vanishing densities (without any assumption on the support of the densities), which is also of interest for image processing applications.…”

Section: Introductionmentioning

confidence: 99%

Multi-physics optimal transportation and image interpolation

2015

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Optimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier [4] where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation.

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Numerical solution of the Optimal Transportation problem using the Monge–Ampère equation

Cited by 192 publications

References 41 publications

A nonlinear principal component decomposition

A nonlinear principal component decomposition

A Sparse Multiscale Algorithm for Dense Optimal Transport

Multi-physics optimal transportation and image interpolation

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