Randomized Wasserstein Barycenter Computation: Resampling with Statistical Guarantees

Heinemann, Florian; Munk, Axel; Zemel, Yoav

doi:10.1137/20m1385263

Cited by 9 publications

(12 citation statements)

References 42 publications

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“…Another method computes continuous barycenters using another way of parameterizing them [31]. Further speedups can be obtained by subsampling the given measures [26] or dimensionality reduction of the support point clouds [28].…”

Section: Introductionmentioning

confidence: 99%

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

Lindheim¹

2022

Preprint

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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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Section: Introductionmentioning

confidence: 99%

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

Lindheim¹

2022

Preprint

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“…The coupled two-marginal formulation can be efficiently simulated with the Sinkhorn algorithm [CD14], see also [CP16;Sch19;PC+19]. Other recent approaches to compute the coupled two marginal formulation can be found in [Yan+21;HMZ22]. The multi-marginal formulation has to our knowledge not previously been simulated except for 3 marginals [Ben+15], but we argue that doing so provides a worthwhile alternative method which is very accurate and achieves finer resolution.…”

Section: Introductionmentioning

confidence: 99%

The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splines

Friesecke¹,

Penka²

2022

Preprint

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“…This work makes advances on both of these matters for a specific version of UOT. This thesis is based on the key results from the three articles Heinemann et al [2022b], and Heinemann et al [2022a] found in the Addenda listed as A, B and C, respectively. Chapter 1 provides a general overview over the theory of OT.…”

Section: Prefacementioning

confidence: 99%

“…Starting with the basic theory of linear programming, it also gives a short overview of a range of computational methods for OT problems and particularly the Wasserstein vii barycenter problem. Chapter 2 is based on the results in Heinemann et al [2022b].…”

Section: Prefacementioning

confidence: 99%

“…it is sufficient for this to understand the Wasserstein deviation of a single measure from its empirical measure to understand the deviation of the plug-in estimator 2 . It holds (see for the original result and Heinemann et al [2022b] for the 1 To streamline presentation results which have been taken from this work are therefore not referenced explicitly in the following section. 2 In the naive upper bound based on the triangle inequality the potentially slower rate of the two measures dominates the speed of convergence.…”

Section: Barycentersmentioning

confidence: 99%

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Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

Heinemann¹

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find the OT map by picking the most cost-efficient pairwise assignment between the points. However, when considering part b), this approach fails. Here, one measure still has eight support points with mass 1/8, but the other one has seven support points with mass 1/7, respectively. It is immediately clear that in this scenario, the set of transport maps is empty. Though, of course it is still possible to transport mass between the two measures. The key ingredient which is missing is the ability to split mass, i.e. for a given location x ∈ supp(µ 1 ), to be allowed to send parts of its mass to different locations in supp(µ 2 ). This does not provide an OT map, however, it leads to finding an Intuitively, the curve (ν t ) t∈[0,1] describes a locally shortest path between µ 1 and µ 2 with respect to the geometry of the metric space (P(Y), W p ). A specific time point ν t can be understood as a W p -interpolation of µ 1 and µ 2 with weight 1 − t for µ 1 and t for µ 2 . Recall the example OT plan from Figure 1.2(b). Following the trajectory of the corresponding geodesic (see Figure 1.3(b)), it can be seen that the ellipse is being squeezed together, while it simultaneously moves from bottom-left to the top-right.This can be seen easily by noting that since ν is a geodesic it holds W p p (µ 1 , ν 0.5 ) + W p p (µ 2 , ν 0.5 ) = 2 1−p W p p (µ 1 , µ 2 ) (1.7)

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Randomized Wasserstein Barycenter Computation: Resampling with Statistical Guarantees

Cited by 9 publications

References 42 publications

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

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

The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splines

Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

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