On the geometry of geodesics in discrete optimal transport

Erbar, Matthias; Maas, Jan; Wirth, Melchior

doi:10.1007/s00526-018-1456-1

Cited by 11 publications

(11 citation statements)

References 23 publications

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“…This result yields a noncommutative version of the dual formula obtained independently by Erbar, Maas and the author [EMW19] and Gangbo, Li and Mou [GLM19] for the Wasserstein-like transport distance on graphs. In fact, we prove a dual formula that is not only valid for the metric W, but also for the entropic regularization recently introduced by Becker-Li [BL21].…”

Section: Introductionsupporting

confidence: 74%

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A dual formula for the noncommutative transport distance

Wirth

2021

Preprint

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Section: Introductionsupporting

confidence: 74%

“…In this section we prove the duality theorem announced in the introduction. Our strategy follows the same lines as the proof in the commutative case in [EMW19]. It crucially relies on the Rockefellar-Fenchel duality theorem quoted below.…”

Section: Dualitymentioning

confidence: 96%

A dual formula for the noncommutative transport distance

Wirth

2021

Preprint

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“…Nevertheless we note that this is a interesting problem. A possible approach in this direction is via duality using nonlocal analogues of the Hamilton-Jacobi equations, similarly to how this problem was recently treated in the discrete setting in [28,30]. Following the work of Erbar [23] we show that the nonlocal Wasserstein quasi-metric generates a topology on the set of probability measures which is stronger than the W 1 topology (i.e., the Monge distance or the 1-Wasserstein metric).…”

Section: Upwind Nonlocal Transportation Metricmentioning

confidence: 85%

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

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“…The focus of this article lies on the noncommutative transport distance introduced in the first approach. More precisely, we prove a dual formula that is a noncommutative analog of the expression of the classical -Wasserstein distance in terms of subsolutions of the Hamilton–Jacobi equation [ 5 , 24 ] This result yields a noncommutative version of the dual formula obtained independently by Erbar et al [ 15 ] and Gangb et al [ 16 ] for the Wasserstein-like transport distance on graphs. In fact, we prove a dual formula that is not only valid for the metric , but also for the entropic regularization recently introduced by Becker–Li [ 3 ].…”

Section: Introductionmentioning

confidence: 85%