Wasserstein barycenters are NP-hard to compute

Altschuler, Jason M.; Boix-Adsera, Enric

doi:10.48550/arxiv.2101.01100

Cited by 3 publications

(7 citation statements)

References 19 publications

(35 reference statements)

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“…7 that the larger the batch, the more concentrated the trajectories of mn,s become. Additionally, Table 3 summarizes the means of the distance W 2 2 to the true model m 0 , using the sequences after t = 100 against the empirical estimator using all the simulations with k ≥ 100. Finally, Table 4 shows the standard deviation of the distance W 2 2 to the true model m 0 , which can be seen to decrease as the batch size grows.…”

Section: Distance Between the Empirical Barycenter And The Bayesian M...mentioning

confidence: 99%

“…We refer the reader to the overview [38] for statistical applications, to the works [12,15,16,41] for applications in machine learning, and to [3,4,10,33] for a presentation of recent developments and further references. As a cautionary tale, we mention that the problem of computing Wasserstein barycenters is known to be NP-hard, see [2]. To cope with this, fast methods aiming at learning and generating approximate Wasserstein barycenters on the basis of neural networks techniques, have also been proposed, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian learning with Wasserstein barycenters

Rios¹,

Backhoff-Veraguas²,

Fontbona³

et al. 2022

ESAIM: PS

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We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasser- stein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection frame- work, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper [7], and provide a numerical example for experimental validation of the proposed method.

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Section: Distance Between the Empirical Barycenter And The Bayesian M...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bayesian learning with Wasserstein barycenters

Rios¹,

Backhoff-Veraguas²,

Fontbona³

et al. 2022

ESAIM: PS

View full text Add to dashboard Cite

show abstract

“…This eliminates the exponential dimension dependence of state-of-the-art convergence rates [Che+20], which aligns with the empirical performance of this algorithm (see Figure 1). It also stands in sharp contrast to the setting of discrete distributions in which there are computational complexity barriers to achieving even polynomial dimension dependence [AB21b].…”

Section: Contributionsmentioning

confidence: 99%

“…The second is that generically, these problems are computationally hard in high dimensions. For instance, Wasserstein barycenters and geometric medians of discrete distributions are NP-hard to compute (even approximately) in high dimension [AB21b].…”

Section: Introductionmentioning

confidence: 99%

Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent

Altschuler¹,

Chewi²,

Gerber³

et al. 2021

Preprint

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We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-theshelf methods such as Euclidean GD and SDP solvers. This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropicallyregularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems.

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“…However, this approach does not scale well with the dimension as noted by Altschuler and Boix-Adserà (2021). Indeed, they showed that computing the infimum of ν → λ i W 2 2 (µ i , ν) when the µ i are discrete measures is a NP-hard problem in the dimension, even when only approximate solutions are acceptable.…”

Section: Introductionmentioning

confidence: 99%

Sampling From the Wasserstein Barycenter

Daaloul,

Gouic,

Liandrat

et al. 2021

Preprint

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This work presents an algorithm to sample from the Wasserstein barycenter of absolutely continuous measures. Our method is based on the gradient flow of the multimarginal formulation of the Wasserstein barycenter, with an additive penalization to account for the marginal constraints. We prove that the minimum of this penalized multimarginal formulation is achieved for a coupling that is close to the Wasserstein barycenter. The performances of the algorithm are showcased in several settings.

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Wasserstein barycenters are NP-hard to compute

Cited by 3 publications

References 19 publications

Bayesian learning with Wasserstein barycenters

Bayesian learning with Wasserstein barycenters

Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent

Sampling From the Wasserstein Barycenter

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