Unbalanced Multi-Marginal Optimal Transport

Beier, Florian; Lindheim, Johannes von; Neumayer, Sebastian; Steidl, Gabriele

doi:10.48550/arxiv.2103.10854

Cited by 9 publications

(18 citation statements)

References 30 publications

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“…This approach was neither considered for gLOT nor for gLGW so far. Further, multimarginals may be addressed, see [6]. Finally, we are interested in further meaningful applications of our approach.…”

Section: Discussionmentioning

confidence: 99%

On a linear Gromov-Wasserstein distance

Beier,

Beinert,

Steidl

2021

Preprint

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Gromov-Wasserstein distances are generalization of Wasserstein distances, which are invariant under certain distance preserving transformations. Although a simplified version of optimal transport in Wasserstein spaces, called linear optimal transport (LOT), was successfully used in certain applications, there does not exist a notation of linear Gromov-Wasserstein distances so far. In this paper, we propose a definition of linear Gromov-Wasserstein distances. We motivate our approach by a generalized LOT model, which is based on barycentric projections. Numerical examples illustrate that the linear Gromov-Wasserstein distances, similarly as LOT, can replace the expensive computation of pairwise Gromov-Wasserstein distances in certain applications.

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Section: Discussionmentioning

confidence: 99%

On a linear Gromov-Wasserstein distance

Beier,

Beinert,

Steidl

2021

Preprint

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“…For another application, we lift the experiment of Section 4.2 from interpolation of measures in Euclidean space to interpolation of textures via the synthesis method from [27], using their publicly available source code 7 . While the authors already interpolated between two different textures in that paper, requiring only the solution of a two-marginal optimal transport problem to obtain a barycenter, we can do this for multiple textures using Algorithm 2.…”

Section: Texture Interpolationmentioning

confidence: 99%

“…It was originally introduced by [23] in the continuous setting for squared Euclidean costs and further generalized in various ways, e.g. to entropy regularized [9,25] and unbalanced variants with non-exact marginal constraints [7]. For a survey with general cost functions and their applications, e.g.…”

Section: Introductionmentioning

confidence: 99%

Approximative Algorithms for Multi-Marginal Optimal Transport and Free-Support Wasserstein Barycenters

Lindheim¹

2022

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Computationally solving multi-marginal optimal transport (MOT) with squared Euclidean costs for N discrete probability measures has recently attracted considerable attention, in part because of the correspondence of its solutions with Wasserstein-2 barycenters, which have many applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. While entropic regularization has been successfully applied to approximate Wasserstein barycenters, this loses the sparsity of the optimal solution, making it difficult to solve the MOT problem directly in practice because of the curse of dimensionality. Thus, for obtaining barycenters, one usually resorts to fixed-support restrictions to a grid, which is, however, prohibitive in higher ambient dimensions d. In this paper, after analyzing the relationship between MOT and barycenters, we present two algorithms to approximate the solution of MOT directly, requiring mainly just N − 1 standard twomarginal OT computations. Thus, they are fast, memory-efficient and easy to implement and can be used with any sparse OT solver as a black box. Moreover, they produce sparse solutions and show promising numerical results. We analyze these algorithms theoretically, proving upper and lower bounds for the relative approximation error.

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“…For more details, we refer to (Koltai et al, 2021). In the case n > 2, we use multi-marginal unbalanced OT, see (Beier et al, 2021). The resulting optimal transport plans πt yield corresponding transfer operators…”

Section: Coherently Evolving Features: Dynamical Spectral Clusteringmentioning

confidence: 99%

Definition, detection, and tracking of persistent structures in atmospheric flows

Lindheim¹,

Harikrishnan²,

Dörffel³

et al. 2021

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Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" -to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence.

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Unbalanced Multi-Marginal Optimal Transport

Cited by 9 publications

References 30 publications

On a linear Gromov-Wasserstein distance

On a linear Gromov-Wasserstein distance

Approximative Algorithms for Multi-Marginal Optimal Transport and Free-Support Wasserstein Barycenters

Definition, detection, and tracking of persistent structures in atmospheric flows

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