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Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm
[article]
2022
arXiv
pre-print
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We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of m discrete probability measures supported on a finite metric space of size ...
Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the FastIBP algorithm in practice. ...
Xi Chen is supported by National Science Foundation via the Grant IIS-1845444. ...
arXiv:2002.04783v11
fatcat:xrcbe5xpujbxhpcfmu5aqna5lu
Fast Discrete Distribution Clustering Using Wasserstein Barycenter with Sparse Support
[article]
2017
arXiv
pre-print
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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. ...
In this paper, we develop a modified Bregman ADMM approach for computing the approximate discrete Wasserstein barycenter of large clusters. ...
The primary computational infrastructures used were supported by the NSF under Grant Nos. ACI-0821527 (CyberStar) and ACI-1053575 (XSEDE). ...
arXiv:1510.00012v4
fatcat:fxe6akdscnhxtkvwu2637mngcq
Bayesian Learning with Wasserstein Barycenters
[article]
2022
arXiv
pre-print
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Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC], and provide ...
We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the ...
and Electronic Engineering FB0008 (FT). ...
arXiv:1805.10833v5
fatcat:rj3hu22myfdk5abvltiwzftn2i
Bayesian learning with Wasserstein barycenters
2022
E S A I M: Probability & Statistics
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Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper, and provide a numerical example for ...
We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the ...
of the paper and refine our results. ...
doi:10.1051/ps/2022015
fatcat:3yjgjfltova5zlcwxcioxxzbuy
Kantorovich-Rubinstein distance and barycenter for finitely supported measures: Foundations and Algorithms
[article]
2022
arXiv
pre-print
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Additionally, we prove the existence of sparse KR barycenters and discuss potential computational approaches. ...
We also consider barycenters based on the recently introduced Gaussian Hellinger-Kantorovich and Wasserstein-Fisher-Rao distances. ...
Heinemann and M. Klatt gratefully acknowledge support from the DFG Research Training Group 2088 Discovering structure in complex data: Statistics meets optimization and inverse problems. A. ...
arXiv:2112.03581v3
fatcat:k5xonaplxbbqjozz5d2lnculw4
Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters
[article]
2022
arXiv
pre-print
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In this work, we propose constructing the nominal distribution in optimal transport-based distributionally robust optimization problems through the notion of Wasserstein barycenter as an aggregation of ...
In many applications in statistics and machine learning, the availability of data samples from multiple possibly heterogeneous sources has become increasingly prevalent. ...
B.2.1 Entropic-Wasserstein and Sinkhorn Barycenters Similar to Wasserstein distances, Wasserstein barycenters are also NP-hard to compute in general (Altschuler and Boix-Adserà, 2022) . ...
arXiv:2203.12136v2
fatcat:2eamrgkmofdhxctvqktyikydeu
Tangential Fixpoint Iterations for Gromov-Wasserstein Barycenters
[article]
2024
arXiv
pre-print
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However, because the computation of GW itself already poses a quadratic and non-convex optimization problem, the determination of GW barycenters is a hard task and algorithms for their computation are ...
The resulting algorithm is capable of producing qualitative shape interpolations between multiple 3d shapes with support sizes of over thousands of points in reasonable time. ...
A remedy to this has been proposed in [11] , where the support of the seeked barycenter is fixed a-priori. ...
arXiv:2403.08612v1
fatcat:fmhlwuifs5e4fkfkmw4gjamz54
Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery
[article]
2022
arXiv
pre-print
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More specifically, we show that a variational formulation of this problem is equivalent to computing a Wasserstein barycenter. ...
We revisit the problem of recovering a low-rank positive semidefinite matrix from rank-one projections using tools from optimal transport. ...
This fixed point iteration forms the basis of our efficient low-rank algorithm. ...
arXiv:2210.14671v1
fatcat:73kvn3jzifgjjhsdjvipeih6pq
Wasserstein K-means for clustering probability distributions
[article]
2022
arXiv
pre-print
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Due to non-negative Alexandrov curvature of the Wasserstein space, barycenters suffer from regularity and non-robustness issues. ...
The peculiar behaviors of Wasserstein barycenters may make the centroid-based formulation fail to represent the within-cluster data points, while the more direct distance-based K-means approach and its ...
Acknowledgments and Disclosure of Funding Xiaohui Chen was partially supported by NSF CAREER grant DMS-1752614. Yun Yang was partially supported by NSF grant DMS-2210717. ...
arXiv:2209.06975v2
fatcat:oa7bksbprrh2bp6s6ti6ehw34e
Computational Optimal Transport: With Applications to Data Science
2019
Foundations and Trends® in Machine Learning
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Gabriel Peyré and Marco Cuturi (2019), "Computational Optimal Transport", Foundations and Trends R in Machine Learning: Vol. 11, No. 5-6, pp 355-607. DOI: 10.1561/2200000073. ...
"A fixed-point approach to barycenters in Wasserstein space". Journal of Mathematical Analysis and Applications. 441(2): 744-762. Alvarez-Melis, D., S. Jegelka, and T. S. Jaakkola. 2018. ...
"Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters". In: Advances in Neural Information Processing Systems 31. Ed. by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. ...
doi:10.1561/2200000073
fatcat:qxumbjeeojf6hkbkaqescsfffy
Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey
2024
IEEE Transactions on Pattern Analysis and Machine Intelligence
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2023 pre-print arXiv:2305.05080v1 fatcat:pn6wfieuljbxhnnukptltourruOther Versions
First, we explain the optimal transport background and introduce different flavors (i.e. mathematical formulations), properties, and notable applications. ...
Optimal Transport (OT) is a mathematical framework that first emerged in the eighteenth century and has led to a plethora of methods for answering many theoretical and applied questions. ...
This work was funded by the CSIRO Machine Learning and Artificial Intelligence Future Science Platform. ...
doi:10.1109/tpami.2024.3379571
pmid:38507387
fatcat:a5bouontvncq5lvq5ulw2gkjwy
Barycentric-alignment and reconstruction loss minimization for domain generalization
[article]
2022
arXiv
pre-print
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Compared to previous bounds, our bound introduces two new terms: (i) the Wasserstein-2 barycenter term for the distribution alignment between domains and (ii) the reconstruction loss term for measuring ...
Based on the new upper bound, we propose a novel DG algorithm that simultaneously minimizes the classification loss, the barycenter loss, and the reconstruction loss. ...
The proposed algorithm requires calculating Wasserstein-2 barycenter and its supporting points. ...
arXiv:2109.01902v5
fatcat:k2m4luw5ybaorp4cqnimjnd6sm
Computational Optimal Transport
[article]
2020
arXiv
pre-print
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Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation ...
) or machine learning (for regression, classification and density fitting). ...
Some of their inputs have shaped this work, and we would like to thank in particular Jean-David Benamou, Yann Brenier, Guillaume Carlier, Vincent Duval and the entire MOKAPLAN team at Inria; Francis Bach ...
arXiv:1803.00567v4
fatcat:zgannw6i6beqde5bx7pj62uyry
Exploration via Hindsight Goal Generation
[article]
2019
arXiv
pre-print
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We have extensively evaluated our goal generation algorithm on a number of robotic manipulation tasks and demonstrated substantially improvement over the original HER in terms of sample efficiency. ...
Hindsight Experience Replay (HER), a recent advance, has greatly improved sample efficiency and practical applicability for such problems. ...
Then we fix π and optimize the the hindsight set T subject to the diversity constraint, which is a variant of the well-known Wasserstein Barycenter problem with a bias term (the value function) for each ...
arXiv:1906.04279v3
fatcat:grdkxsolojhlxa7nujmexyvo2u
A Unified Framework for Implicit Sinkhorn Differentiation
[article]
2022
arXiv
pre-print
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The Sinkhorn operator has recently experienced a surge of popularity in computer vision and related fields. One major reason is its ease of integration into deep learning frameworks. ...
To allow for an efficient training of respective neural networks, we propose an algorithm that obtains analytical gradients of a Sinkhorn layer via implicit differentiation. ...
Acknowledgements This work was supported by the ERC Advanced Grant SIMULACRON and the Munich School for Data Science. ...
arXiv:2205.06688v1
fatcat:fmry7nunn5f4vlxipwuychgdne
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