Scaling algorithms for unbalanced optimal transport problems

Chizat, Lénaïc; Peyré, Gabriel; Schmitzer, Bernhard; Vialard, François-Xavier

doi:10.1090/mcom/3303

Cited by 292 publications

(396 citation statements)

References 76 publications

(104 reference statements)

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“…This scheme was extended in [5] to unbalanced transport problem and used in [4] to compute Cournot-Nash equilibria. It is well known that the solution of (5.2) is of the form γ i,j = a i (η ε ) i,j b j , where a, b ∈ R N and η ε ∈ R N ×N .…”

Section: Numerical Simulationsmentioning

confidence: 99%

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On the Wasserstein distance between mutually singular measures

Buttazzo

Carlier

Laborde

2018

Advances in Calculus of Variations

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We study the Wasserstein distance between two measures µ, ν which are mutually singular. In particular, we are interested in minimization problems of the formwhere µ is a given probability and A is contained in the class µ ⊥ of probabilities that are singular with respect to µ. Several cases for A are considered; in particular, when A consists of L 1 densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problemwhere k > 0 is a fixed constant, P (A) is the perimeter of A, and both sets A, B may vary.

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Section: Numerical Simulationsmentioning

confidence: 99%

“…We refer to [10,5] for the convergence of this algorithm to a solution of (5.2). The advantage of this method is that computing prox KL G l can be done easily.…”

Section: Numerical Simulationsmentioning

confidence: 99%

On the Wasserstein distance between mutually singular measures

Buttazzo

Carlier

Laborde

2018

Advances in Calculus of Variations

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“…Yet more recently, we learned from the reviewer that active research on the thresholding technique is currently underway with new manuscripts emerging while we approached the final revision of this paper. In particular, log domain scaling or more sophisticated schemes have been proposed to stabilize the low ε case for Sinkhorn algorithm [33], [34]. At extra computational costs, these new methods produce sharper coupling results than the standard IBP does.…”

Section: Comparison Between B-admm and Iterative Bregman Projectionmentioning

confidence: 99%

Fast Discrete Distribution Clustering Using Wasserstein Barycenter With Sparse Support

Wang

et al. 2017

IEEE Trans. Signal Process.

101

133

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Abstract-In a variety of research areas, the weighted bag of vectors and the histogram are widely used descriptors for complex objects. Both can be expressed as discrete distributions. D2-clustering pursues the minimum total within-cluster variation for a set of discrete distributions subject to the KantorovichWasserstein metric. D2-clustering has a severe scalability issue, the bottleneck being the computation of a centroid distribution, called Wasserstein barycenter, that minimizes its sum of squared distances to the cluster members. In this paper, we develop a modified Bregman ADMM approach for computing the approximate discrete Wasserstein barycenter of large clusters. In the case when the support points of the barycenters are unknown and have low cardinality, our method achieves high accuracy empirically at a much reduced computational cost. The strengths and weaknesses of our method and its alternatives are examined through experiments, and we recommend scenarios for their respective usage. Moreover, we develop both serial and parallelized versions of the algorithm. By experimenting with large-scale data, we demonstrate the computational efficiency of the new methods and investigate their convergence properties and numerical stability. The clustering results obtained on several datasets in different domains are highly competitive in comparison with some widely used methods in the corresponding areas.

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“…Recent applications include astronomy [9, 18, 19], biomedical sciences [3, 25–27, 77, 81, 82, 88, 89], colour transfer [14, 17, 49, 62, 63], computer vision and graphics [7, 44, 60, 65, 68, 74, 75], imaging [36, 40, 64], information theory [78], machine learning [1, 15, 20, 34, 37, 48, 76], operational research [69] and signal processing [54, 58]. …”

Section: Introductionmentioning

confidence: 99%

A Transportation $$L^p$$ Distance for Signal Analysis

et al. 2017

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Transport based distances, such as the Wasserstein distance and earth mover'sdistance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the TLp distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled non-positive one-dimensional signals and two-dimensional images, and color transfer.

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Scaling algorithms for unbalanced optimal transport problems

Cited by 292 publications

References 76 publications

On the Wasserstein distance between mutually singular measures

On the Wasserstein distance between mutually singular measures

Fast Discrete Distribution Clustering Using Wasserstein Barycenter With Sparse Support

A Transportation $$L^p$$ Distance for Signal Analysis

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