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

Ba, Fatima Antarou; Quellmalz, Michael

doi:10.3390/a15090311

Cited by 6 publications

(7 citation statements)

References 65 publications

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“…where the coefficients c jk are drawn from the interval [5,20] and functions g jk,1 , g jk,2 are drawn from the set…”

Section: Manifold Optimization On So(d) For Jointly Sparsifying a Set...mentioning

confidence: 99%

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Sparse additive function decompositions facing basis transforms

Antarou Ba,

Melnyk,

Wald

et al. 2024

FoDS

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High-dimensional real-world systems can often be well characterized by a small number of simultaneous low-complexity interactions. The analysis of variance (ANOVA) decomposition and the anchored decomposition are typical techniques to find sparse additive decompositions of functions. In this paper, we are interested in a setting, where these decompositions are not directly sparse, but become so after an appropriate basis transform. Noting that the sparsity of those additive function decompositions is equivalent to the fact that most of its mixed partial derivatives vanish, we can exploit a connection to the underlying function graphs to determine an orthogonal transform that realizes the appropriate basis change. This is done in three steps: we apply singular value decomposition to minimize the number of vertices of the function graph, and joint block diagonalization techniques of families of matrices followed by sparse minimization based on relaxations of the zero "norm" for minimizing the number of edges. For the latter one, we propose and analyze minimization techniques over the manifold of special orthogonal matrices. Various numerical examples illustrate the reliability of our approach for functions having, after a basis transform, a sparse additive decomposition into summands with at most two variables.

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“…where the coefficients c jk are drawn from the interval [5,20] and functions g jk,1 , g jk,2 are drawn from the set…”

Section: Manifold Optimization On So(d) For Jointly Sparsifying a Set...mentioning

confidence: 99%

“…Recently, we dealt with such decompositions in the context of multimarginal optimal transport, where only special structured cost functions can be treated in an efficient way [5,8].…”

mentioning

confidence: 99%

Sparse additive function decompositions facing basis transforms

Antarou Ba,

Melnyk,

Wald

et al. 2024

FoDS

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“…λ −j n := (−1) j λ j n , and λ j n = 0 otherwise. Here Y k n denote the spherical harmonics (7) and D k,j n the rotational harmonics (13). Moreover, there are constants C 1 , C 2 > 0 such that…”

Section: Normalized Semicircle Transform Of Functionsmentioning

confidence: 99%

“…As a reference, we consider the spherical 2-Wasserstein barycenter (2) and its entropyregularized counterpart [62], whose computation with the Sinkhorn algorithm [46] can be implemented efficiently, see [7,14]. We apply the Python optimal transport library [28] for both, where the Sinkhorn algorithm uses the regularization parameter 0.01 and a maximum number of 1000 iterations.…”

Section: Interpolation Between Mises-fisher Distributionsmentioning

confidence: 99%

Sliced optimal transport on the sphere

Quellmalz,

Beinert,

Steidl

2023

Inverse Problems

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Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform, which analytically yields the solutions of the reduced transport problems. Inspired by this concept, we propose two adaptions for optimal transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we introduce the vertical slice transform, which integrates along all circles orthogonal to a given direction. Secondly, we introduce a semicircle transform, which integrates along all half great circles with an appropriate weight function. Both transforms are generalized to arbitrary measures on the sphere. While the vertical slice transform can be combined with optimal transport on the interval and leads to a sliced Wasserstein distance restricted to even probability measures, the semicircle transform is related to optimal transport on the circle and results in a different sliced Wasserstein distance for arbitrary probability measures. The applicability of both novel sliced optimal transport concepts on the sphere is demonstrated by proof-of-concept examples dealing with the interpolation and classification of spherical probability measures. The numerical implementation relies on the singular value decompositions of both transforms and fast Fourier techniques. For the inversion with respect to probability measures, we propose the minimization of an entropy-regularized Kullback-Leibler divergence, which can be numerically realized using a primal-dual proximal splitting algorithm.

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

Cited by 6 publications

References 65 publications

Sparse additive function decompositions facing basis transforms

Sparse additive function decompositions facing basis transforms

Sliced optimal transport on the sphere

Soft Quantization Using Entropic Regularization

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