DOTmark – A Benchmark for Discrete Optimal Transport

Schrieber, Jörn; Schuhmacher, Dominic; Gottschlich, Carsten

doi:10.1109/access.2016.2639065

Cited by 28 publications

(23 citation statements)

References 24 publications

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“…Quantitative evaluation of OT solvers. For discrete OT methods, a benchmark dataset [35] exists but the mechanism for producing the dataset does not extend to continuous OT. Existing continuous solvers are typically evaluated on a set of self-generated examples or tested in generative models without evaluating its actual OT performance.…”

Section: Background On Optimal Transportmentioning

confidence: 99%

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Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark

Korotin¹,

Li²,

Genevay³

et al. 2021

Preprint

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Despite the recent popularity of neural network-based solvers for optimal transport (OT), there is no standard quantitative way to evaluate their performance. In this paper, we address this issue for quadratic-cost transport-specifically, computation of the Wasserstein-2 distance, a commonly-used formulation of optimal transport in machine learning. To overcome the challenge of computing ground truth transport maps between continuous measures needed to assess these solvers, we use inputconvex neural networks (ICNN) to construct pairs of measures whose ground truth OT maps can be obtained analytically. This strategy yields pairs of continuous benchmark measures in high-dimensional spaces such as spaces of images. We thoroughly evaluate existing optimal transport solvers using these benchmark measures. Even though these solvers perform well in downstream tasks, many do not faithfully recover optimal transport maps. To investigate the cause of this discrepancy, we further test the solvers in a setting of image generation. Our study reveals crucial limitations of existing solvers and shows that increased OT accuracy does not necessarily correlate to better results downstream.Solving optimal transport (OT) with continuous methods has become widespread in machine learning, including methods for large-scale OT [11,36] and the popular Wasserstein Generative Adversarial Network (W-GAN) [3,12]. Rather than discretizing the problem [31], continuous OT algorithms use neural networks or kernel expansions to estimate transport maps or dual solutions. This helps scale OT to large-scale and higher-dimensional problems not handled by discrete methods. Notable successes of continuous OT are in generative modeling [42,20,19,7] and domain adaptation [43,37,25].In these applications, OT is typically incorporated as part of the loss terms for a neural network model. For example, in W-GANs, the OT cost is used as a loss function for the generator; the model incorporates a neural network-based OT solver to estimate the loss. Although recent W-GANs provide state-of-the-art generative performance, however, it remains unclear to which extent this success is connected to OT. For example, [28,32,38] show that popular solvers for the Wasserstein-1 (W 1 ) distance in GANs fail to estimate W 1 accurately. While W-GANs were initially introduced Preprint.

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Section: Background On Optimal Transportmentioning

confidence: 99%

“…Our benchmark measures can be used to evaluate future W 2 solvers in high-dimensional spaces, a crucial step to improve the transparency and replicability of continuous OT research. Note the benchmark from [35] does not fulfill this purpose, since it is designed to test discrete OT methods and uses discrete low-dimensional measures with limited support.…”

mentioning

confidence: 99%

Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark

Korotin¹,

Li²,

Genevay³

et al. 2021

Preprint

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“…The linear programming problem (ILPP) is formulated by an assignment matrix ( , ℎ) (Das et al 2020) where ℎ >> and each EV is assigned to a PFCS. This matrix is grounded on the renowned assignment problem (Aktel et al 2017) with the combination of traffic flow structure (Schrieber et al 2017). Since, traffic flow is a dynamic event, hence, it is expressed as ( , ℎ, ), where, t ∈ .…”

Section: ) Assignment Matrix For Ilpmentioning

confidence: 99%

Allocation of EV Public Charging Station in Renewable based Distribution Network using HHO Considering Uncertainties and Traffic Congestion

Pal¹,

Bhattacharya²,

Chakraborty³

2021

Preprint

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Electric vehicle (EV) is the growing vehicular technology for sustainable development to reduce carbon emission and to save fossil fuel. The charging station (CS) is necessary at appropriate locations to facilitate the EV owners to charge their vehicle as well as to keep the distribution system parameters within permissible limits. Besides that, the selection of a charging station is also a significant task for the EV user to reduce battery energy wastage while reaching the EV charging station. This paper presents a realistic solution for the allocation of public fast-charging stations (PFCS) along with solar distributed generation (SDG). A 33 node radial distribution network is superimposed with the corresponding traffic network to allocate PFCSs and SDGs. Two interconnected stages of optimization are used in this work. The first part deals with the optimization of PFCS’s locations and SDG’s locations with sizes, to minimize the energy loss and to improve voltage profile using harris hawk optimization (HHO) and few other soft computing techniques. The second part handles the proper assignment of EVs to the PFCSs with less consumption of the EV’s energy considering the road distances with traffic congestion using linear programming (LP), where the shortest paths are decided by Dijkstra's algorithm. The 2m point estimation method (2m PEM) is employed to handle the uncertainties associated with EVs and SDGs. The robustness of solutions are tested using wilcoxon signed rank test and quade test.

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“…In addition to a variance parameter, which we kept fixed, the Matérn covariance function has parameters for the scale γ of the correlations, which we varied among 0.05, 0.15 and 0.5, and the smoothness s of the generated surface, which we varied between 0.5 and 2.5 corresponding to a continuous surface and a C 2 -surface, respectively. The simulation mechanism is similar to the ones for classes 2-5 in the benchmark DOTmark proposed in Schrieber et al (2017), but allows to investigate the influence of individual parameters more directly. Figure 3 shows one realization for each parameter combination.…”

Section: Performance Evaluationmentioning

confidence: 99%

Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case

Hartmann

Schuhmacher

2020

Math Meth Oper Res

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We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al.

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DOTmark – A Benchmark for Discrete Optimal Transport

Cited by 28 publications

References 24 publications

Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark

Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark

Allocation of EV Public Charging Station in Renewable based Distribution Network using HHO Considering Uncertainties and Traffic Congestion

Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case

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