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Journal articles on the topic 'Wasserstein barycenters'

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Author: Grafiati

Published: 19 October 2024

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1

Chi, Jinjin, Zhiyao Yang, Ximing Li, Jihong Ouyang, and Renchu Guan. "Variational Wasserstein Barycenters with C-cyclical Monotonicity Regularization." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 6 (June 26, 2023): 7157–65. http://dx.doi.org/10.1609/aaai.v37i6.25873.

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Wasserstein barycenter, built on the theory of Optimal Transport (OT), provides a powerful framework to aggregate probability distributions, and it has increasingly attracted great attention within the machine learning community. However, it is often intractable to precisely compute, especially for high dimensional and continuous settings. To alleviate this problem, we develop a novel regularization by using the fact that c-cyclical monotonicity is often necessary and sufficient conditions for optimality in OT problems, and incorporate it into the dual formulation of Wasserstein barycenters. For efficient computations, we adopt a variational distribution as the approximation of the true continuous barycenter, so as to frame the Wasserstein barycenters problem as an optimization problem with respect to variational parameters. Upon those ideas, we propose a novel end-to-end continuous approximation method, namely Variational Wasserstein Barycenters with c-Cyclical Monotonicity Regularization (VWB-CMR), given sample access to the input distributions. We show theoretical convergence analysis and demonstrate the superior performance of VWB-CMR on synthetic data and real applications of subset posterior aggregation.

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Xu, Hongteng, Dixin Luo, Lawrence Carin, and Hongyuan Zha. "Learning Graphons via Structured Gromov-Wasserstein Barycenters." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 12 (May 18, 2021): 10505–13. http://dx.doi.org/10.1609/aaai.v35i12.17257.

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We propose a novel and principled method to learn a nonparametric graph model called graphon, which is defined in an infinite-dimensional space and represents arbitrary-size graphs. Based on the weak regularity lemma from the theory of graphons, we leverage a step function to approximate a graphon. We show that the cut distance of graphons can be relaxed to the Gromov-Wasserstein distance of their step functions. Accordingly, given a set of graphs generated by an underlying graphon, we learn the corresponding step function as the Gromov-Wasserstein barycenter of the given graphs. Furthermore, we develop several enhancements and extensions of the basic algorithm, e.g., the smoothed Gromov-Wasserstein barycenter for guaranteeing the continuity of the learned graphons and the mixed Gromov-Wasserstein barycenters for learning multiple structured graphons. The proposed approach overcomes drawbacks of prior state-of-the-art methods, and outperforms them on both synthetic and real-world data. The code is available at https://github.com/HongtengXu/SGWB-Graphon.

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Bigot, Jérémie, and Thierry Klein. "Characterization of barycenters in the Wasserstein space by averaging optimal transport maps." ESAIM: Probability and Statistics 22 (2018): 35–57. http://dx.doi.org/10.1051/ps/2017020.

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This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

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4

Agueh, Martial, and Guillaume Carlier. "Barycenters in the Wasserstein Space." SIAM Journal on Mathematical Analysis 43, no. 2 (January 2011): 904–24. http://dx.doi.org/10.1137/100805741.

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5

Kim, Young-Heon, and Brendan Pass. "Wasserstein barycenters over Riemannian manifolds." Advances in Mathematics 307 (February 2017): 640–83. http://dx.doi.org/10.1016/j.aim.2016.11.026.

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6

Baum, Marcus, Peter Willett, and Uwe D. Hanebeck. "On Wasserstein Barycenters and MMOSPA Estimation." IEEE Signal Processing Letters 22, no. 10 (October 2015): 1511–15. http://dx.doi.org/10.1109/lsp.2015.2410217.

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7

Puccetti, Giovanni, Ludger Rüschendorf, and Steven Vanduffel. "On the computation of Wasserstein barycenters." Journal of Multivariate Analysis 176 (March 2020): 104581. http://dx.doi.org/10.1016/j.jmva.2019.104581.

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8

Le Gouic, Thibaut, and Jean-Michel Loubes. "Existence and consistency of Wasserstein barycenters." Probability Theory and Related Fields 168, no. 3-4 (August 17, 2016): 901–17. http://dx.doi.org/10.1007/s00440-016-0727-z.

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9

Buzun, Nazar. "Gaussian Approximation for Penalized Wasserstein Barycenters." Mathematical Methods of Statistics 32, no. 1 (March 2023): 1–26. http://dx.doi.org/10.3103/s1066530723010039.

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10

Sow, Babacar, Rodolphe Le Riche, Julien Pelamatti, Merlin Keller, and Sanaa Zannane. "Wasserstein-Based Evolutionary Operators for Optimizing Sets of Points: Application to Wind-Farm Layout Design." Applied Sciences 14, no. 17 (September 5, 2024): 7916. http://dx.doi.org/10.3390/app14177916.

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This paper introduces an evolutionary algorithm for objective functions defined over clouds of points of varying sizes. Such design variables are modeled as uniform discrete measures with finite support and the crossover and mutation operators of the algorithm are defined using the Wasserstein barycenter. We prove that the Wasserstein-based crossover has a contracting property in the sense that the support of the generated measure is included in the closed convex hull of the union of the two parents’ supports. We introduce boundary mutations to counteract this contraction. Variants of evolutionary operators based on Wasserstein barycenters are studied. We compare the resulting algorithm to a more classical, sequence-based, evolutionary algorithm on a family of test functions that include a wind-farm layout problem. The results show that Wasserstein-based evolutionary operators better capture the underlying geometrical structures of the considered test functions and outperform a reference evolutionary algorithm in the vast majority of the cases. The tests indicate that the mutation operators play a major part in the performances of the algorithms.

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11

Battisti, Beatrice, Tobias Blickhan, Guillaume Enchery, Virginie Ehrlacher, Damiano Lombardi, and Olga Mula. "Wasserstein model reduction approach for parametrized flow problems in porous media." ESAIM: Proceedings and Surveys 73 (2023): 28–47. http://dx.doi.org/10.1051/proc/202373028.

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The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

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12

Altschuler, Jason M., and Enric Boix-Adserà. "Wasserstein Barycenters Are NP-Hard to Compute." SIAM Journal on Mathematics of Data Science 4, no. 1 (February 10, 2022): 179–203. http://dx.doi.org/10.1137/21m1390062.

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13

Bonneel, Nicolas, Julien Rabin, Gabriel Peyré, and Hanspeter Pfister. "Sliced and Radon Wasserstein Barycenters of Measures." Journal of Mathematical Imaging and Vision 51, no. 1 (April 8, 2014): 22–45. http://dx.doi.org/10.1007/s10851-014-0506-3.

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14

Simou, Effrosyni, Dorina Thanou, and Pascal Frossard. "node2coords: Graph Representation Learning with Wasserstein Barycenters." IEEE Transactions on Signal and Information Processing over Networks 7 (2021): 17–29. http://dx.doi.org/10.1109/tsipn.2020.3041940.

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15

Bigot, Jérémie, Elsa Cazelles, and Nicolas Papadakis. "Penalization of Barycenters in the Wasserstein Space." SIAM Journal on Mathematical Analysis 51, no. 3 (January 2019): 2261–85. http://dx.doi.org/10.1137/18m1185065.

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Bigot, Jérémie, Elsa Cazelles, and Nicolas Papadakis. "Data-driven regularization of Wasserstein barycenters with an application to multivariate density registration." Information and Inference: A Journal of the IMA 8, no. 4 (November 30, 2019): 719–55. http://dx.doi.org/10.1093/imaiai/iaz023.

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Abstract We present a framework to simultaneously align and smoothen data in the form of multiple point clouds sampled from unknown densities with support in a $d$-dimensional Euclidean space. This work is motivated by applications in bioinformatics where researchers aim to automatically homogenize large datasets to compare and analyze characteristics within a same cell population. Inconveniently, the information acquired is most certainly noisy due to misalignment caused by technical variations of the environment. To overcome this problem, we propose to register multiple point clouds by using the notion of regularized barycenters (or Fréchet mean) of a set of probability measures with respect to the Wasserstein metric. The first approach consists in penalizing a Wasserstein barycenter with a convex functional as recently proposed in [5]. The second strategy is to transform the Wasserstein metric itself into an entropy regularized transportation cost between probability measures as introduced in [12]. The main contribution of this work is to propose data-driven choices for the regularization parameters involved in each approach using the Goldenshluger–Lepski’s principle. Simulated data sampled from Gaussian mixtures are used to illustrate each method, and an application to the analysis of flow cytometry data is finally proposed. This way of choosing of the regularization parameter for the Sinkhorn barycenter is also analyzed through the prism of an oracle inequality that relates the error made by such data-driven estimators to the one of an ideal estimator.

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17

Xu, Hongtengl. "Gromov-Wasserstein Factorization Models for Graph Clustering." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 6478–85. http://dx.doi.org/10.1609/aaai.v34i04.6120.

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We propose a new nonlinear factorization model for graphs that are with topological structures, and optionally, node attributes. This model is based on a pseudometric called Gromov-Wasserstein (GW) discrepancy, which compares graphs in a relational way. It estimates observed graphs as GW barycenters constructed by a set of atoms with different weights. By minimizing the GW discrepancy between each observed graph and its GW barycenter-based estimation, we learn the atoms and their weights associated with the observed graphs. The model achieves a novel and flexible factorization mechanism under GW discrepancy, in which both the observed graphs and the learnable atoms can be unaligned and with different sizes. We design an effective approximate algorithm for learning this Gromov-Wasserstein factorization (GWF) model, unrolling loopy computations as stacked modules and computing gradients with backpropagation. The stacked modules can be with two different architectures, which correspond to the proximal point algorithm (PPA) and Bregman alternating direction method of multipliers (BADMM), respectively. Experiments show that our model obtains encouraging results on clustering graphs.

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18

Carlier, Guillaume, Katharina Eichinger, and Alexey Kroshnin. "Entropic-Wasserstein Barycenters: PDE Characterization, Regularity, and CLT." SIAM Journal on Mathematical Analysis 53, no. 5 (January 2021): 5880–914. http://dx.doi.org/10.1137/20m1387262.

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19

Carlier, Guillaume, Katharina Eichinger, and Alexey Kroshnin. "Entropic-Wasserstein Barycenters: PDE Characterization, Regularity, and CLT." SIAM Journal on Mathematical Analysis 53, no. 5 (January 2021): 5880–914. http://dx.doi.org/10.1137/20m1387262.

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20

Pont, Mathieu, Jules Vidal, Julie Delon, and Julien Tierny. "Wasserstein Distances, Geodesics and Barycenters of Merge Trees." IEEE Transactions on Visualization and Computer Graphics 28, no. 1 (January 2022): 291–301. http://dx.doi.org/10.1109/tvcg.2021.3114839.

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21

Anderes, Ethan, Steffen Borgwardt, and Jacob Miller. "Discrete Wasserstein barycenters: optimal transport for discrete data." Mathematical Methods of Operations Research 84, no. 2 (June 16, 2016): 389–409. http://dx.doi.org/10.1007/s00186-016-0549-x.

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22

Jiang, Yin. "Absolute Continuity of Wasserstein Barycenters Over Alexandrov Spaces." Canadian Journal of Mathematics 69, no. 5 (October 1, 2017): 1087–108. http://dx.doi.org/10.4153/cjm-2016-035-8.

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AbstractIn this paper, we prove that on a compact, n-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures μ1 ,… , μm is absolutely continuous with respect to the n-dimensional Hausdorff measure if one of them is.

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23

Xiang, Yue, Dixin Luo, and Hongteng Xu. "Privacy-Preserved Evolutionary Graph Modeling via Gromov-Wasserstein Autoregression." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 12 (June 26, 2023): 14566–74. http://dx.doi.org/10.1609/aaai.v37i12.26703.

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Real-world graphs like social networks are often evolutionary over time, whose observations at different timestamps lead to graph sequences. Modeling such evolutionary graphs is important for many applications, but solving this problem often requires the correspondence between the graphs at different timestamps, which may leak private node information, e.g., the temporal behavior patterns of the nodes. We proposed a Gromov-Wasserstein Autoregressive (GWAR) model to capture the generative mechanisms of evolutionary graphs, which does not require the correspondence information and thus preserves the privacy of the graphs' nodes. This model consists of two autoregressions, predicting the number of nodes and the probabilities of nodes and edges, respectively. The model takes observed graphs as its input and predicts future graphs via solving a joint graph alignment and merging task. This task leads to a fused Gromov-Wasserstein (FGW) barycenter problem, in which we approximate the alignment of the graphs based on a novel inductive fused Gromov-Wasserstein (IFGW) distance. The IFGW distance is parameterized by neural networks and can be learned under mild assumptions, thus, we can infer the FGW barycenters without iterative optimization and predict future graphs efficiently. Experiments show that our GWAR achieves encouraging performance in modeling evolutionary graphs in privacy-preserving scenarios.

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24

Carlier, Guillaume, Adam Oberman, and Edouard Oudet. "Numerical methods for matching for teams and Wasserstein barycenters." ESAIM: Mathematical Modelling and Numerical Analysis 49, no. 6 (November 2015): 1621–42. http://dx.doi.org/10.1051/m2an/2015033.

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25

Álvarez-Esteban, Pedro C., E. del Barrio, J. A. Cuesta-Albertos, and C. Matrán. "A fixed-point approach to barycenters in Wasserstein space." Journal of Mathematical Analysis and Applications 441, no. 2 (September 2016): 744–62. http://dx.doi.org/10.1016/j.jmaa.2016.04.045.

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26

Gao, Tingran, Shahab Asoodeh, Yi Huang, and James Evans. "Wasserstein Soft Label Propagation on Hypergraphs: Algorithm and Generalization Error Bounds." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3630–37. http://dx.doi.org/10.1609/aaai.v33i01.33013630.

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Inspired by recent interests of developing machine learning and data mining algorithms on hypergraphs, we investigate in this paper the semi-supervised learning algorithm of propagating ”soft labels” (e.g. probability distributions, class membership scores) over hypergraphs, by means of optimal transportation. Borrowing insights from Wasserstein propagation on graphs [Solomon et al. 2014], we re-formulate the label propagation procedure as a message-passing algorithm, which renders itself naturally to a generalization applicable to hypergraphs through Wasserstein barycenters. Furthermore, in a PAC learning framework, we provide generalization error bounds for propagating one-dimensional distributions on graphs and hypergraphs using 2-Wasserstein distance, by establishing the algorithmic stability of the proposed semisupervised learning algorithm. These theoretical results also shed new lights upon deeper understandings of the Wasserstein propagation on graphs.

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27

Amari, Shun-ichi, Ryo Karakida, Masafumi Oizumi, and Marco Cuturi. "Information Geometry for Regularized Optimal Transport and Barycenters of Patterns." Neural Computation 31, no. 5 (May 2019): 827–48. http://dx.doi.org/10.1162/neco_a_01178.

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We propose a new divergence on the manifold of probability distributions, building on the entropic regularization of optimal transportation problems. As Cuturi ( 2013 ) showed, regularizing the optimal transport problem with an entropic term is known to bring several computational benefits. However, because of that regularization, the resulting approximation of the optimal transport cost does not define a proper distance or divergence between probability distributions. We recently tried to introduce a family of divergences connecting the Wasserstein distance and the Kullback-Leibler divergence from an information geometry point of view (see Amari, Karakida, & Oizumi, 2018 ). However, that proposal was not able to retain key intuitive aspects of the Wasserstein geometry, such as translation invariance, which plays a key role when used in the more general problem of computing optimal transport barycenters. The divergence we propose in this work is able to retain such properties and admits an intuitive interpretation.

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Ponti, Andrea, Ilaria Giordani, Antonio Candelieri, and Francesco Archetti. "Wasserstein-Enabled Leaks Localization in Water Distribution Networks." Water 16, no. 3 (January 27, 2024): 412. http://dx.doi.org/10.3390/w16030412.

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Leaks in water distribution networks are estimated to account for up to 30% of the total distributed water; moreover, the increasing demand and the skyrocketing energy cost have made leak localization and adoption ever more important to water utilities. Each leak scenario is run on a simulation model to compute the resulting values of pressure and flows over the whole network. The values recorded by the sensors are seen as features of one leak scenario and can be considered as the signature of the leak. The key distinguishing element in this paper is to consider the entire distribution of data, representing a leak as a probability distribution. In this representation, the similarity between leaks can be captured by the Wasserstein distance. This choice matches the physics of the system as follows: the equations modeling the generation of flow and pressure data are non-linear. The signatures obtained through the simulation of a set of leak scenarios are non-linearly clustered in the Wasserstein space using Wasserstein barycenters as centroids. As a new set of measurements arrives, its signature is associated with the cluster with the closest barycenter. The location of the simulated leaks belonging to that cluster are the possible locations of the observed leak. This new framework allows a richer representation of pressure and flow data embedding both the modeling and the computational modules in a space whose elements are discrete probability distribution endowed with the Wasserstein distance. Experiments on benchmark and real-world networks confirm the feasibility of the proposed approach.

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29

Nobari, Elham, and Bijan Ahmadi Kakavandi. "Wasserstein barycenters in the manifold of all positive definite matrices." Quarterly of Applied Mathematics 77, no. 3 (February 7, 2019): 655–69. http://dx.doi.org/10.1090/qam/1535.

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30

Salaün, Corentin, Iliyan Georgiev, Hans-Peter Seidel, and Gurprit Singh. "Scalable Multi-Class Sampling via Filtered Sliced Optimal Transport." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–14. http://dx.doi.org/10.1145/3550454.3555484.

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We propose a multi-class point optimization formulation based on continuous Wasserstein barycenters. Our formulation is designed to handle hundreds to thousands of optimization objectives and comes with a practical optimization scheme. We demonstrate the effectiveness of our framework on various sampling applications like stippling, object placement, and Monte-Carlo integration. We a derive multi-class error bound for perceptual rendering error which can be minimized using our optimization. We provide source code at https://github.com/iribis/filtered-sliced-optimal-transport.

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31

Larvaron, Benjamin, Marianne Clausel, Antoine Bertoncello, Sébastien Benjamin, Georges Oppenheim, and Clément Bertin. "Conditional Wasserstein barycenters to predict battery health degradation at unobserved experimental conditions." Journal of Energy Storage 78 (February 2024): 110015. http://dx.doi.org/10.1016/j.est.2023.110015.

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32

Likmeta, Amarildo, Matteo Sacco, Alberto Maria Metelli, and Marcello Restelli. "Wasserstein Actor-Critic: Directed Exploration via Optimism for Continuous-Actions Control." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 7 (June 26, 2023): 8782–90. http://dx.doi.org/10.1609/aaai.v37i7.26056.

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Uncertainty quantification has been extensively used as a means to achieve efficient directed exploration in Reinforcement Learning (RL). However, state-of-the-art methods for continuous actions still suffer from high sample complexity requirements. Indeed, they either completely lack strategies for propagating the epistemic uncertainty throughout the updates, or they mix it with aleatoric uncertainty while learning the full return distribution (e.g., distributional RL). In this paper, we propose Wasserstein Actor-Critic (WAC), an actor-critic architecture inspired by the recent Wasserstein Q-Learning (WQL), that employs approximate Q-posteriors to represent the epistemic uncertainty and Wasserstein barycenters for uncertainty propagation across the state-action space. WAC enforces exploration in a principled way by guiding the policy learning process with the optimization of an upper bound of the Q-value estimates. Furthermore, we study some peculiar issues that arise when using function approximation, coupled with the uncertainty estimation, and propose a regularized loss for the uncertainty estimation. Finally, we evaluate our algorithm on standard MujoCo tasks as well as suite of continuous-actions domains, where exploration is crucial, in comparison with state-of-the-art baselines. Additional details and results can be found in the supplementary material with our Arxiv preprint.

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Li, Jia, and Fuqing Zhang. "Geometry-Sensitive Ensemble Mean Based on Wasserstein Barycenters: Proof-of-Concept on Cloud Simulations." Journal of Computational and Graphical Statistics 27, no. 4 (October 2, 2018): 785–97. http://dx.doi.org/10.1080/10618600.2018.1448831.

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34

Friesecke, Gero, and Maximilian Penka. "The GenCol Algorithm for High-Dimensional Optimal Transport: General Formulation and Application to Barycenters and Wasserstein Splines." SIAM Journal on Mathematics of Data Science 5, no. 4 (October 25, 2023): 899–919. http://dx.doi.org/10.1137/22m1524254.

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35

Papayiannis, G. I., G. N. Domazakis, D. Drivaliaris, S. Koukoulas, A. E. Tsekrekos, and A. N. Yannacopoulos. "On clustering uncertain and structured data with Wasserstein barycenters and a geodesic criterion for the number of clusters." Journal of Statistical Computation and Simulation 91, no. 13 (March 30, 2021): 2569–94. http://dx.doi.org/10.1080/00949655.2021.1903463.

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36

Ehrlacher, Virginie, Damiano Lombardi, Olga Mula, and François-Xavier Vialard. "Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 6 (November 2020): 2159–97. http://dx.doi.org/10.1051/m2an/2020013.

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We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. Since the notion of linear vectorial spaces does not exist in general metric spaces, both approaches result in nonlinear approximation methods. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the L2-Wasserstein space.

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Ecoffet, Arthur, Geoffrey Woollard, Artem Kushner, Frédéric Poitevin, and Khanh Dao Duc. "Application of transport-based metric for continuous interpolation between cryo-EM density maps." AIMS Mathematics 7, no. 1 (2021): 986–99. http://dx.doi.org/10.3934/math.2022059.

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<abstract><p>Cryogenic electron microscopy (cryo-EM) has become widely used for the past few years in structural biology, to collect single images of macromolecules "frozen in time". As this technique facilitates the identification of multiple conformational states adopted by the same molecule, a direct product of it is a set of 3D volumes, also called EM maps. To gain more insights on the possible mechanisms that govern transitions between different states, and hence the mode of action of a molecule, we recently introduced a bioinformatic tool that interpolates and generates morphing trajectories joining two given EM maps. This tool is based on recent advances made in optimal transport, that allow efficient evaluation of Wasserstein barycenters of 3D shapes. As the overall performance of the method depends on various key parameters, including the sensitivity of the regularization parameter, we performed various numerical experiments to demonstrate how MorphOT can be applied in different contexts and settings. Finally, we discuss current limitations and further potential connections between other optimal transport theories and the conformational heterogeneity problem inherent with cryo-EM data.</p></abstract>

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Hu, Francois, Philipp Ratz, and Arthur Charpentier. "A Sequentially Fair Mechanism for Multiple Sensitive Attributes." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 11 (March 24, 2024): 12502–10. http://dx.doi.org/10.1609/aaai.v38i11.29143.

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In the standard use case of Algorithmic Fairness, the goal is to eliminate the relationship between a sensitive variable and a corresponding score. Throughout recent years, the scientific community has developed a host of definitions and tools to solve this task, which work well in many practical applications. However, the applicability and effectivity of these tools and definitions becomes less straightfoward in the case of multiple sensitive attributes. To tackle this issue, we propose a sequential framework, which allows to progressively achieve fairness across a set of sensitive features. We accomplish this by leveraging multi-marginal Wasserstein barycenters, which extends the standard notion of Strong Demographic Parity to the case with multiple sensitive characteristics. This method also provides a closed-form solution for the optimal, sequentially fair predictor, permitting a clear interpretation of inter-sensitive feature correlations. Our approach seamlessly extends to approximate fairness, enveloping a framework accommodating the trade-off between risk and unfairness. This extension permits a targeted prioritization of fairness improvements for a specific attribute within a set of sensitive attributes, allowing for a case specific adaptation. A data-driven estimation procedure for the derived solution is developed, and comprehensive numerical experiments are conducted on both synthetic and real datasets. Our empirical findings decisively underscore the practical efficacy of our post-processing approach in fostering fair decision-making.

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Kim, Young-Heon, and Brendan Pass. "A Canonical Barycenter via Wasserstein Regularization." SIAM Journal on Mathematical Analysis 50, no. 2 (January 2018): 1817–28. http://dx.doi.org/10.1137/17m1123055.

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40

Beier, Florian, Robert Beinert, and Gabriele Steidl. "Multi-marginal Gromov–Wasserstein transport and barycentres." Information and Inference: A Journal of the IMA 12, no. 4 (September 18, 2023): 2720–52. http://dx.doi.org/10.1093/imaiai/iaad041.

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Abstract Gromov–Wasserstein (GW) distances are combinations of Gromov–Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis. In this paper, we introduce the concept of multi-marginal GW transport between a set of mm-spaces as well as its regularized and unbalanced versions. As a special case, we discuss multi-marginal fused variants, which combine the structure information of an mm-space with label information from an additional label space. To tackle the new formulations numerically, we consider the bi-convex relaxation of the multi-marginal GW problem, which is tight in the balanced case if the cost function is conditionally negative definite. The relaxed model can be solved by an alternating minimization, where each step can be performed by a multi-marginal Sinkhorn scheme. We show relations of our multi-marginal GW problem to (unbalanced, fused) GW barycentres and present various numerical results, which indicate the potential of the concept.

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41

Jin, Cong, Junhao Wang, Jin Wei, Lifeng Tan, Shouxun Liu, Wei Zhao, Shan Liu, and Xin Lv. "Multimedia Analysis and Fusion via Wasserstein Barycenter." International Journal of Networked and Distributed Computing 8, no. 2 (2020): 58. http://dx.doi.org/10.2991/ijndc.k.200217.001.

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Arias-Serna, M. Andrea, Jean Michel Loubes, and Francisco J. Caro-Lopera. "Multi-Variate Risk Measures under Wasserstein Barycenter." Risks 10, no. 9 (September 7, 2022): 180. http://dx.doi.org/10.3390/risks10090180.

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When the uni-variate risk measure analysis is generalized into the multi-variate setting, many complex theoretical and applied problems arise, and therefore the mathematical models used for risk quantification usually present model risk. As a result, regulators have started to require that the internal models used by financial institutions are more precise. For this task, we propose a novel multi-variate risk measure, based on the notion of the Wasserstein barycenter. The proposed approach robustly characterizes the company’s exposure, filtering the partial information available from individual sources into an aggregate risk measure, providing an easily computable estimation of the total risk incurred. The new approach allows effective computation of Wasserstein barycenter risk measures in any location–scatter family, including the Gaussian case. In such cases, the Wasserstein barycenter Value-at-Risk belongs to the same family, thus it is characterized just by its mean and deviation. It is important to highlight that the proposed risk measure is expressed in closed analytic forms which facilitate its use in day-to-day risk management. The performance of the new multi-variate risk measures is illustrated in United States market indices of high volatility during the global financial crisis (2008) and during the COVID-19 pandemic situation, showing that the proposed approach provides the best forecasts of risk measures not only for “normal periods”, but also for periods of high volatility.

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Heinemann, Florian, Axel Munk, and Yoav Zemel. "Randomized Wasserstein Barycenter Computation: Resampling with Statistical Guarantees." SIAM Journal on Mathematics of Data Science 4, no. 1 (February 28, 2022): 229–59. http://dx.doi.org/10.1137/20m1385263.

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Kim, Young-Heon, and Brendan Pass. "Nonpositive curvature, the variance functional, and the Wasserstein barycenter." Proceedings of the American Mathematical Society 148, no. 4 (January 13, 2020): 1745–56. http://dx.doi.org/10.1090/proc/14840.

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Eustasio Del Barrio, Jean-Michel Loubes, and Bruno Pelletier. "An Inverse Problem: Recovery of a Distribution Using Wasserstein Barycenter." Annals of Economics and Statistics, no. 128 (2017): 229. http://dx.doi.org/10.15609/annaeconstat2009.128.0229.

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Borgwardt, Steffen, and Stephan Patterson. "On the computational complexity of finding a sparse Wasserstein barycenter." Journal of Combinatorial Optimization 41, no. 3 (March 3, 2021): 736–61. http://dx.doi.org/10.1007/s10878-021-00713-5.

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Ye, Jianbo, Panruo Wu, James Z. Wang, and Jia Li. "Fast Discrete Distribution Clustering Using Wasserstein Barycenter With Sparse Support." IEEE Transactions on Signal Processing 65, no. 9 (May 1, 2017): 2317–32. http://dx.doi.org/10.1109/tsp.2017.2659647.

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Cumings-Menon, Ryan, and Minchul Shin. "Probability Forecast Combination via Entropy Regularized Wasserstein Distance." Entropy 22, no. 9 (August 25, 2020): 929. http://dx.doi.org/10.3390/e22090929.

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We propose probability and density forecast combination methods that are defined using the entropy regularized Wasserstein distance. First, we provide a theoretical characterization of the combined density forecast based on the regularized Wasserstein distance under the assumption. More specifically, we show that the regularized Wasserstein barycenter between multivariate Gaussian input densities is multivariate Gaussian, and provide a simple way to compute mean and its variance–covariance matrix. Second, we show how this type of regularization can improve the predictive power of the resulting combined density. Third, we provide a method for choosing the tuning parameter that governs the strength of regularization. Lastly, we apply our proposed method to the U.S. inflation rate density forecasting, and illustrate how the entropy regularization can improve the quality of predictive density relative to its unregularized counterpart.

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Bigot, Jérémie. "Statistical data analysis in the Wasserstein space." ESAIM: Proceedings and Surveys 68 (2020): 1–19. http://dx.doi.org/10.1051/proc/202068001.

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This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on the use of Wasserstein distances and tools from optimal transport to analyse such data. In particular, we highlight the benefits of using the notions of barycenter and geodesic PCA in the Wasserstein space for the purpose of learning the principal modes of geometric variation in a dataset. In this setting, we discuss existing works and we present some research perspectives related to the emerging field of statistical optimal transport.

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Bocquet, Marc, Pierre J. Vanderbecken, Alban Farchi, Joffrey Dumont Le Brazidec, and Yelva Roustan. "Bridging classical data assimilation and optimal transport: the 3D-Var case." Nonlinear Processes in Geophysics 31, no. 3 (July 12, 2024): 335–57. http://dx.doi.org/10.5194/npg-31-335-2024.

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Abstract. Because optimal transport (OT) acts as displacement interpolation in physical space rather than as interpolation in value space, it can avoid double-penalty errors generated by mislocations of geophysical fields. As such, it provides a very attractive metric for non-negative, sharp field comparison – the Wasserstein distance – which could further be used in data assimilation (DA) for the geosciences. However, the algorithmic and numerical implementations of such a distance are not straightforward. Moreover, its theoretical formulation within typical DA problems faces conceptual challenges, resulting in scarce contributions on the topic in the literature. We formulate the problem in a way that offers a unified view with respect to both classical DA and OT. The resulting OTDA framework accounts for both the classical source of prior errors, background and observation, and a Wasserstein barycentre in between states which are pre-images of the background state and observation vector. We show that the hybrid OTDA analysis can be decomposed as a simpler OTDA problem involving a single Wasserstein distance, followed by a Wasserstein barycentre problem that ignores the prior errors and can be seen as a McCann interpolant. We also propose a less enlightening but straightforward solution to the full OTDA problem, which includes the derivation of its analysis error covariance matrix. Thanks to these theoretical developments, we are able to extend the classical 3D-Var/BLUE (best linear unbiased estimator) paradigm at the core of most classical DA schemes. The resulting formalism is very flexible and can account for sparse, noisy observations and non-Gaussian error statistics. It is illustrated by simple one- and two-dimensional examples that show the richness of the new types of analysis offered by this unification.

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