Gromov–Hausdorff limit of Wasserstein spaces on point clouds

Trillos, Nicolás García

doi:10.1007/s00526-020-1729-3

Cited by 8 publications

(5 citation statements)

References 30 publications

(32 reference statements)

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“…A natural question is whether the space (P(S),W d ) converges to (P(M),W 2 ) as S becomes a finer and finer discretization of a manifold M. For a discrete Wasserstein distance like the one of Maas [2011], based on the graph structure of S-which corresponds to the case where velocity fields are discretized by their values on edges and a particular choice of scalar product-the answer is known to be positive in the case where M is the flat torus [Gigli and Maas 2013;Trillos 2017] in the sense of Gromov-Hausdorff convergence of metric spaces, while a very recent work by Gladbach, Kopfer and Mass [2018] has refined the analysis and exhibits necessary conditions for such a convergence to hold. The high technicality of the proofs of these results, however, indicates that the question for our particular definition is likely to be challenging and out of the scope of the present article.…”

Section: Riemannian Structure Of the Space Of Probabilities On A Disc...mentioning

confidence: 99%

Dynamical optimal transport on discrete surfaces

2018

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Fig. 1. Given two probability distributions over a discrete surface (left and right), our algorithm generates an interpolation that takes the geometric structure of the surface into account.We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.

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Section: Riemannian Structure Of the Space Of Probabilities On A Disc...mentioning

confidence: 99%

Dynamical optimal transport on discrete surfaces

2018

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“…Remark 1.4 (Convergence on geometric graphs). A convergence result for discrete transport distances on a large class of geometric graphs associated to point clouds on the d-dimensional torus has been obtained in [GT17]. This result applies in particular to iid points sampled from the uniform distribution on the torus.…”

Section: Introductionmentioning

confidence: 98%

Homogenisation of one-dimensional discrete optimal transport

Gladbach

Kopfer

Maas

et al. 2020

Journal de Mathématiques Pures et Appliquées

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This paper deals with dynamical optimal transport metrics defined by discretisation of the Benamou-Benamou formula for the Kantorovich metric W 2 . Such metrics appear naturally in discretisations of W 2 -gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to W 2 , unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the nonlocal mobility at the discrete level. Loosely speaking, the result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.We start by informally introducing the main objects of study in this paper and present the main result. For more formal definitions we refer to Section 2 below.Continuous optimal transport. Let P(S) (resp. M (S)) denote the set of Borel probability measures (resp. signed measures) on a Polish space (S, d). We will work on the one-dimensional torus S 1 = R/Z and use the convention that arithmetic operations are understood modulo 1.

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“…The large scale behaviour of optimal transport on random point clouds has been studied by Garcia-Trillos, who proved convergence to the Wasserstein distance [Gar20].…”

Section: Introductionmentioning

confidence: 99%

Homogenisation of dynamical optimal transport on periodic graphs

Gladbach¹,

Kopfer²,

Maas³

et al. 2021

Preprint

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This paper deals with the large-scale behaviour of dynamical optimal transport on Z d -periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density.Our homogenisation result is derived from a Γ-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs. Contents 42 9. Analysis of the cell problem 57 Appendix A. The Kantorovich-Rubinstein metric on signed measures 66 Appendix B. Norms on curves in the space of measures 68 Appendix C. Domain property of convex functions 68 Appendix D. Notation 68 References 70

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Gromov–Hausdorff limit of Wasserstein spaces on point clouds

Cited by 8 publications

References 30 publications

Dynamical optimal transport on discrete surfaces

Dynamical optimal transport on discrete surfaces

Homogenisation of one-dimensional discrete optimal transport

Homogenisation of dynamical optimal transport on periodic graphs

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