From Optimal Transport to Discrepancy

Neumayer, Sebastian; Steidl, Gabriele

doi:10.1007/978-3-030-03009-4_95-1

Cited by 8 publications

(3 citation statements)

References 32 publications

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“…From (3.1a) and the arguments below, we infer that for λ → ∞, we obtain the standard Wasserstein distance in the limit. We refer to [24], where the regularization parameter is studied. The constructive line of research by [11] affirms that when the regularization parameter λ is not sufficiently large, the transportation plan and the regularized Wasserstein distance may be incompatible.…”

Section: Etna Kent State University and Johann Radon Institute (Ricam)mentioning

confidence: 99%

Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm

Lakshmanan¹,

Pichler²,

Potts³

2023

etna

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This contribution features an accelerated computation of Sinkhorn's algorithm, which approximates the Wasserstein transportation distance, by employing nonequispaced fast Fourier transforms (NFFT). The proposed algorithm allows approximations of the Wasserstein distance by involving not more than O(n log n) operations for probability measures supported by n points. Furthermore, the proposed method avoids expensive allocations of the characterizing matrices. With this numerical acceleration, the transportation distance is accessible to probability measures out of reach so far. Numerical experiments using synthetic and real data affirm the computational advantage and superiority.

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Section: Etna Kent State University and Johann Radon Institute (Ricam)mentioning

confidence: 99%

Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm

Lakshmanan¹,

Pichler²,

Potts³

2023

etna

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“…As an avenue for constructive research, the above-cited study presented a multitude of results aimed at gaining a comprehensive understanding of the subtleties involved in enhancing the computational performance of entropy-optimal transport (cf. Ramdas et al [18], Neumayer and Steidl [19], Altschuler et al [20], Lakshmanan et al [21], Ba and Quellmalz [22], Lakshmanan and Pichler [23]). These findings have served as a valuable foundation for further exploration in the field of optimal transport, providing insights into both the intricacies of the topic and potential avenues for improvement.…”

Section: Related Work and Contributionsmentioning

confidence: 99%

Soft Quantization Using Entropic Regularization

Lakshmanan,

Pichler

2023

Entropy

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The quantization problem aims to find the best possible approximation of probability measures on Rd using finite and discrete measures. The Wasserstein distance is a typical choice to measure the quality of the approximation. This contribution investigates the properties and robustness of the entropy-regularized quantization problem, which relaxes the standard quantization problem. The proposed approximation technique naturally adopts the softmin function, which is well known for its robustness from both theoretical and practicability standpoints. Moreover, we use the entropy-regularized Wasserstein distance to evaluate the quality of the soft quantization problem’s approximation, and we implement a stochastic gradient approach to achieve the optimal solutions. The control parameter in our proposed method allows for the adjustment of the optimization problem’s difficulty level, providing significant advantages when dealing with exceptionally challenging problems of interest. As well, this contribution empirically illustrates the performance of the method in various expositions.

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“…Since the numerical computation of a transport plan is difficult in general, a regularization term such as the entropy [10,29], Kullback-Leibler divergence [40], general fdivergence [50] or L 2 -regularization [13,37] can be added to make the problem strictly convex. Different approaches such as the Sinkhorn algorithm [29,43], stochastic gradient descent [28], the Gauss-Seidel method [37], or the proximal splitting [5] have been used to iteratively determine a minimizing sequence of the MOT problem.…”

Section: Introductionmentioning

confidence: 99%

Accelerating the Sinkhorn Algorithm for Sparse Multi-Marginal Optimal Transport via Fast Fourier Transforms

Quellmalz

2022

Algorithms

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We consider the numerical solution of the discrete multi-marginal optimal transport (MOT) by means of the Sinkhorn algorithm. In general, the Sinkhorn algorithm suffers from the curse of dimensionality with respect to the number of marginals. If the MOT cost function decouples according to a tree or circle, its complexity is linear in the number of marginal measures. In this case, we speed up the convolution with the radial kernel required in the Sinkhorn algorithm via non-uniform fast Fourier methods. Each step of the proposed accelerated Sinkhorn algorithm with a tree-structured cost function has a complexity of O(KN) instead of the classical O(KN2) for straightforward matrix–vector operations, where K is the number of marginals and each marginal measure is supported on, at most, N points. In the case of a circle-structured cost function, the complexity improves from O(KN3) to O(KN2). This is confirmed through numerical experiments.

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From Optimal Transport to Discrepancy

Cited by 8 publications

References 32 publications

Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm

Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm

Soft Quantization Using Entropic Regularization

Accelerating the Sinkhorn Algorithm for Sparse Multi-Marginal Optimal Transport via Fast Fourier Transforms

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