An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows

2021 ◽  
Author(s):  
Alessio Figalli ◽  
Federico Glaudo
Keyword(s):  
Optimal Transport ◽  
Gradient Flows ◽  
Wasserstein Distances

scholarly journals The back-and-forth method for Wasserstein gradient flows

2021 ◽  
Vol 27 ◽  
pp. 28
Author(s):  
Matt Jacobs ◽  
Wonjun Lee ◽  
Flavien Léger
Keyword(s):  
Large Class ◽  
Large Scale ◽  
Dual Problem ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Gradient Flows ◽  
Primal Problem ◽  
Transport Problems ◽  
Wasserstein Gradient Flows ◽  
Numer Math

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in Jacobs and Léger [Numer. Math. 146 (2020) 513–544.]. to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.

scholarly journals Statistical data analysis in the Wasserstein space

2020 ◽  
Vol 68 ◽  
pp. 1-19
Author(s):  
Jérémie Bigot
Keyword(s):  
Optimal Transport ◽  
Point Clouds ◽  
Data Set ◽  
Inference Problems ◽  
Research Perspectives ◽  
Random Probability ◽  
Wasserstein Space ◽  
Geometric Variation ◽  
Random Probability Measures ◽  
Wasserstein Distances

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.

scholarly journals Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

2017 ◽  
Vol 49 (2) ◽  
pp. 1385-1418 ◽  
Author(s):  
Guillaume Carlier ◽  
Vincent Duval ◽  
Gabriel Peyré ◽  
Bernhard Schmitzer
Keyword(s):  
Optimal Transport ◽  
Gradient Flows

scholarly journals Tropical optimal transport and Wasserstein distances

2021 ◽  
Author(s):  
Wonjun Lee ◽  
Wuchen Li ◽  
Bo Lin ◽  
Anthea Monod
Keyword(s):  
Phylogenetic Trees ◽  
Optimal Transport ◽  
Tropical Geometry ◽  
Inner Product ◽  
Efficient Computation ◽  
Statistical Framework ◽  
Geometric Setting ◽  
Geometric Space ◽  
Theoretical Foundations ◽  
Wasserstein Distances

AbstractWe study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of $$p=1,2$$ p = 1 , 2 in detail. The case of $$p=1$$ p = 1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $$p=2$$ p = 2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.

scholarly journals Robust Wasserstein profile inference and applications to machine learning

2019 ◽  
Vol 56 (3) ◽  
pp. 830-857 ◽  
Author(s):  
Jose Blanchet ◽  
Yang Kang ◽  
Karthyek Murthy
Keyword(s):  
Machine Learning ◽  
Cross Validation ◽  
Optimal Transport ◽  
Optimization Problems ◽  
Empirical Distribution ◽  
Transport Costs ◽  
Regularization Parameters ◽  
Out Of Sample ◽  
Distributionally Robust ◽  
Wasserstein Distances

AbstractWe show that several machine learning estimators, including square-root least absolute shrinkage and selection and regularized logistic regression, can be represented as solutions to distributionally robust optimization problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (robust Wasserstein profile inference), a novel inference methodology which extends the use of methods inspired by empirical likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.

Spectral analysis of Morse-Smale gradient flows

2019 ◽  
Vol 52 (6) ◽  
pp. 1403-1458
Author(s):  
Nguyen Viet DANG ◽  
Gabriel RIVIERE
Keyword(s):  
Spectral Analysis ◽  
Gradient Flows

scholarly journals Optimal transport and unitary orbits in C⁎-algebras

2021 ◽  
Vol 281 (5) ◽  
pp. 109068
Author(s):  
Bhishan Jacelon ◽  
Karen R. Strung ◽  
Alessandro Vignati
Keyword(s):  
Optimal Transport ◽  
C Algebras
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