Wasserstein Barycenters Are NP-Hard to Compute

Altschuler, Jason M.; Boix-Adserà, Enric

doi:10.1137/21m1390062

Cited by 14 publications

(9 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We call the algorithms above "Greedy" and "Reference" below and compute ν = (M λ ) # π of the resulting multi-marginal transport π (without parallelization). 2 Further, we compute the barycenter using publicly available implementations for the methods [29,24,34], called "Debiased", "IBP", "Product", "MAAIPM" and "Frank-Wolfe" below, 3 the exact barycenter method from [3] called "Exact" below, 4 and the method from [32] called "FastIBP" below. 5 We also tried the BADMM 6 method from [51], but since it did not converge properly, we do not consider it further.…”

Section: Ellipse Datasetmentioning

confidence: 99%

See 1 more Smart Citation

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

Lindheim¹

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Ellipse Datasetmentioning

confidence: 99%

“…Unfortunately, MOT and Wasserstein barycenters are in general hard to compute [4]. Although there are polynomial-time methods for fixed dimension d [3], there is still a need for fast ap-proximations.…”

Section: Introductionmentioning

confidence: 99%

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

Lindheim¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…For both of these choices, we will discuss upper and lower bounds on the relative error Ψ(ν)/Ψ(ν). In general, there is no polynomial-time algorithm that will achieve an error arbitrarily close to 1 with high probability [3].…”

Section: Algorithms For Barycenter Approximationmentioning

confidence: 99%

“…Unfortunately, Wasserstein barycenters and MOT are in general hard to compute [3]. Many algorithms restrict the support of the solution to a fixed set and minimize only over the weights.…”

Section: Introductionmentioning

confidence: 99%

Simple Approximative Algorithms for Free-Support Wasserstein Barycenters

Lindheim¹

2022

Preprint

View full text Add to dashboard Cite

Computing Wasserstein barycenters of discrete measures has recently attracted considerable attention due to its wide variety of applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. In this paper, we analyze two straightforward algorithms for approximating barycenters, which produce sparse support solutions and show promising numerical results. These algorithms require N − 1 and N (N − 1)/2 standard two-marginal OT computations between the N input measures, respectively, so that they are fast, in particular the first algorithm, as well as memory-efficient and easy to implement. Further, they can be used with any OT solver as a black box. Based on relations of the barycenter problem to the multi-marginal optimal transport problem, which are interesting on their own, we prove sharp upper bounds for the relative approximation error. In the second algorithm, this upper bound can be evaluated specifically for the given problem, which always guaranteed an error of at most a few percent in our numerical experiments.

show abstract

“…However, determining these point remains a difficult combinatorial problem. In fact, the problem of finding the sparsest p-Wasserstein barycenter is NP-hard [Altschuler and Boix-Adsera, 2022].…”

Section: Wasserstein Barycentersmentioning

confidence: 99%

Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

Heinemann¹

View full text Add to dashboard Cite

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)

show abstract

Wasserstein Barycenters Are NP-Hard to Compute

Cited by 14 publications

References 18 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

Simple Approximative Algorithms for Free-Support Wasserstein Barycenters

Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

Contact Info

Product

Resources

About