Hardness results for Multimarginal Optimal Transport problems

Altschuler, Jason M.; Boix-Adserà, Enric

doi:10.1016/j.disopt.2021.100669

Cited by 17 publications

(15 citation statements)

References 18 publications

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“…While not fixed-support approaches, the Frank-Wolfe algorithm of [22] and the Functional Gradient Descent algorithm of [28] also suffer from the same two issues. 2 Recent work has shown that in any fixed dimension d, Wasserstein barycenters can in fact be computed exactly in poly(n, k, log U ) time [4]. However, the runtime dependence on dimension is still exponential: for non-constant d, the runtime is (nk) d ⋅ poly(n, k, log U ).…”

Section: Previous Workmentioning

confidence: 99%

“…For further discussion about the algorithmic complexity of MOT, see the recent papers [2,3] and the references within.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…The starting point of our reduction is a well-known equivalent reformulation of the barycenter problem (1.1) as a Multimarginal Optimal Transport (MOT) problem. The reason we use such a reformulation is that it lets us use the recently developed machinery of [2] for proving hardness of structured MOT problems. The precise MOT reformulation we state below is a slightly nonstandard variant that is convenient in the sequel.…”

Section: Step 1: Reducing Multimarginal Optimal Transport To Barycentersmentioning

confidence: 99%

“…Here, we leverage the recent results of [2] to reduce (approximately) computing the minimum entry min ⃗ j∈[n] k C ⃗ j of the cost tensor C, to (approximately) computing the optimal value OPT MOT of the MOT problem with cost tensor C. The upshot is that min ⃗ j∈[n] k C ⃗ j is a combinatorial optimization problem that is phrased in a more amenable way for proving NP-hardness. This in large part due to the following simple geometric interpretation of min ⃗ j∈[n] k C ⃗ j : find k points, one from the support of each µ i , such that their variance is minimized.…”

Section: Step 2: Reducing Finding K Maximally-correlated Vectors To M...mentioning

confidence: 99%

“…The specializations of the exact and approximate reductions in of [2] to the present setting are formally stated as follows. Below, the MOT BARY problem is: given measures µ 1 , .…”

Section: Step 2: Reducing Finding K Maximally-correlated Vectors To M...mentioning

confidence: 99%

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Wasserstein barycenters are NP-hard to compute

Altschuler,

Boix-Adsera

2021

Preprint

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The problem of computing Wasserstein barycenters (a.k.a. Optimal Transport barycenters) has attracted considerable recent attention due to many applications in data science. While there exist polynomial-time algorithms in any fixed dimension, all known runtimes suffer exponentially in the dimension. It is an open question whether this exponential dependence is improvable to a polynomial dependence. This paper proves that unless P = NP, the answer is no. This uncovers a "curse of dimensionality" for Wasserstein barycenter computation which does not occur for Optimal Transport computation. Moreover, our hardness results for computing Wasserstein barycenters extend to approximate computation, to seemingly simple cases of the problem, and to averaging probability distributions in other Optimal Transport metrics.

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Section: Previous Workmentioning

confidence: 99%

“…For further discussion about the algorithmic complexity of MOT, see the recent papers [2,3] and the references within.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Section: Step 1: Reducing Multimarginal Optimal Transport To Barycentersmentioning

confidence: 99%

Section: Step 2: Reducing Finding K Maximally-correlated Vectors To M...mentioning

confidence: 99%

“…The specializations of the exact and approximate reductions in of [2] to the present setting are formally stated as follows. Below, the MOT BARY problem is: given measures µ 1 , .…”

Section: Step 2: Reducing Finding K Maximally-correlated Vectors To M...mentioning

confidence: 99%

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Wasserstein barycenters are NP-hard to compute

Altschuler,

Boix-Adsera

2021

Preprint

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Density functionals based on the mathematical structure of the strong‐interaction limit of DFT

Vuckovic

Gerolin

Daas

et al. 2022

WIREs Comput Mol Sci

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While in principle exact, Kohn-Sham density functional theory-the workhorse of computational chemistry-must rely on approximations for the exchange-correlation functional. Despite staggering successes, present-day approximations still struggle when the effects of electron-electron correlation play a prominent role. The limit in which the electronic Coulomb repulsion completely dominates the exchange-correlation functional offers a well-defined mathematical framework that provides insight for new approximations able to deal with strong correlation. In particular, the mathematical structure of this limit, which is now well-established thanks to its reformulation as an optimal transport problem, points to the use of very different ingredients (or features) with respect to the traditional ones used in present approximations. We focus on strategies to use these new ingredients to build approximations for computational chemistry and highlight future promising directions.

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