Multi-marginal optimal transport using partial information with applications in robust localization and sensor fusion

Elvander, Filip; Haasler, Isabel; Jakobsson, Andreas; Karlsson, Johan

doi:10.1016/j.sigpro.2020.107474

Cited by 55 publications

(54 citation statements)

References 40 publications

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“…This topic has been extensively studied, see, e.g., the monograph [66] and references therein. An extension of this problem is the multi-marginal optimal transport problem, in which a minimum-cost transport plan between several distributions is sought [5], [24], [31], [49], [54], [60], [61]. In this work we consider the discrete case of the latter, where the marginal distributions are given by a finite set of T nonnegative vectors 1 µ1, .…”

Section: A the Graph-strucutre Multi-marginal Optimal Transport Problemmentioning

confidence: 99%

“…This perturbed problem can be solved by using the so-called Sinkhorn iterations. 2 The approach has been extended to the multi-marginal setting [5], [24], [49], however in this case it only partly alleviates the computational difficulty. More precisely, in the multi-marginal setting the entropy term is defined 3 as…”

Section: A the Graph-strucutre Multi-marginal Optimal Transport Problemmentioning

confidence: 99%

“…Problem (2) with a cost tensor structured according to (3) is called a (entropy-regularized) graph-structured multi-marginal optimal transport problem [34], [35]. Moreover, for many graph structures, the projections Pj(M) can be efficiently computed, see, e.g, [1], [5], [24], [33]- [37], [62], and hence the Sinkhorn iterations can be used to efficiently solve such problems.…”

Section: A the Graph-strucutre Multi-marginal Optimal Transport Problemmentioning

confidence: 99%

“…Therefore, from this perspective Algorithm 1 is an "abstract algorithm". Nevertheless, for many graph structures it is possible to compute the projections efficiently by sequentially eliminating the modes of the tensor, see [1], [5], [24], [33]- [37], [62].…”

Section: B Coordinate Ascent Iterations For Solving the Dual Problemmentioning

confidence: 99%

“…, T − 1. The transition of the agents from time j to time j + 1 can thus be specified by the bimarginal projection of the tensor onto the joint two marginals j and j + 1, and thus this problem is a structured tensor problem with structure corresponding to a path graph (see, e.g., [20], [24], [34]). However, when the agents have heterogeneous dynamics or objectives, the distribution at a given time does not contain all necessary information about the past and thus the Markov property does not hold.…”

Section: Introductionmentioning

confidence: 99%

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Graph-structured tensor optimization for nonlinear density control and mean field games

Ringh¹,

Haasler²,

Chen³

et al. 2021

Preprint

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In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problems. This type of problems, which can be seen as a generalization of multi-marginal optimal transport problems with graph-structured costs, appear in many applications. In particular, we show that it can be used to model and solve nonlinear density control problems, including convex dynamic network flow problems and multi-species potential mean field games. The method is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures we derive efficient methods for computing these projections. In particular, these graph structures are the ones that occur in convex dynamic network flow problems and multi-species potential mean field games. We also illustrate the methodology on numerical examples from these problem classes.

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Section: A the Graph-strucutre Multi-marginal Optimal Transport Problemmentioning

confidence: 99%

Section: A the Graph-strucutre Multi-marginal Optimal Transport Problemmentioning

confidence: 99%

Section: A the Graph-strucutre Multi-marginal Optimal Transport Problemmentioning

confidence: 99%

Section: B Coordinate Ascent Iterations For Solving the Dual Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Graph-structured tensor optimization for nonlinear density control and mean field games

Ringh¹,

Haasler²,

Chen³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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Multi‐marginal Approximation of the Linear Gromov–Wasserstein Distance

Beier

Beinert

2023

Proc Appl Math and Mech

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Recently, two concepts from optimal transport theory have successfully been brought to the Gromov–Wasserstein (GW) setting. This introduces a linear version of the GW distance and multi‐marginal GW transport. The former can reduce the computational complexity when computing all GW distances of a large set of inputs. The latter allows for a simultaneous matching of more than two marginals, which can for example be used to compute GW barycenters. The aim of this paper is to show an approximation result which characterizes the linear version as a limit of a multi‐marginal GW formulation.

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Gromov–Wasserstein Transfer Operators

Beier

2023

Lecture Notes in Computer Science

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Multi-marginal optimal transport using partial information with applications in robust localization and sensor fusion

Cited by 55 publications

References 40 publications

Graph-structured tensor optimization for nonlinear density control and mean field games

Graph-structured tensor optimization for nonlinear density control and mean field games

Multi‐marginal Approximation of the Linear Gromov–Wasserstein Distance

Gromov–Wasserstein Transfer Operators

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