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A novel notion of barycenter for probability distributions based on optimal weak mass transport
Elsa Cazelles · Felipe Tobar · Joaquin Fontbona

Tue Dec 07 08:30 AM -- 10:00 AM (PST) @ Virtual

We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlan et al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.

Author Information

Elsa Cazelles (IRIT, CNRS, Université de Toulouse)
Felipe Tobar (Universidad de Chile)

Felipe Tobar is an Assistant Professor at the Data & AI Initiative at Universidad de Chile. He holds Researcher positions at the Center for Mathematical Modeling and the Advanced Center for Electrical Engineering. Felipe received the BSc/MSc degrees in Electrical Engineering (U. de Chile, 2010) and a PhD in Signal Processing (Imperial College London, 2014), and he was an Associate Researcher in Machine Learning at the University of Cambridge (2014-2015). Felipe teaches Statistics and Machine Learning courses at undergraduate, graduate and professional levels. His research interests lie in the interface between Machine Learning and Statistical Signal Processing, including Gaussian processes, spectral estimation, approximate inference, Bayesian nonparametrics, and optimal transport.

Joaquin Fontbona (University of Chile)

Joaquin Fontbona Torres • Born: 4 October 1974 at Santiago de Chile. * Chilean. • Since 2011: Associated Professor, Department of Mathematical Engineering and Center for Mathematical Modeling CMM, UMI- 2807 UChile-CNRS. University of Chile. - 2003-2011: Assistant Professor, Department of Mathematical Engineering, University of Chile. I. EDUCATION: • Graduate Studies: Ph.D. in Mathematics, Universit ́e de Paris 6, 1999-2004. • Undergraduate studies: Mathematical Engineering. Universidad de Chile, 1993-1999. II. RESEARCH: • Research interests: Probability theory, stochastic processes, stochastic modeling with applications in physics, finance, biology, risk modeling, signal processing, mining. • Publications: 1. “Dynamics of a planar Coulomb gas” with F. Bolley and D.Chafai. arxiv.org/abs/1706.08776. Submitted. 2. “Skeletal stochastic differential equations for continuous-state branching process”, with D. Fekete and A. E. Kyprianou. arxiv.org/abs/1702.03533. 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Journal of Applied Probability, 47 (2010), No. 2, 543-561. 23. “The limiting move-to-front search-cost in law of large numbers asymptotic regimes”, with J.Barrera. Annals of Applied Probability 20, (2010), No. 2, 722-755. 24. “Measurability of optimal transportation and strong coupling of martingale mea- sures”, with H. Gu ́erin and S. M ́el ́eard. Electronic Communications in Probability. No. 15, (2010) 124-133. 25. “ Measurability of optimal transportation and convergence rate for Landau type in- teracting particle systems”, with H. Gu ́erin and S. M ́el ́eard. Probability Theory and Related Fields 143, No. 3-4, (2009), 329–351. 26. “ On prolific individuals in a supercritical continuous state branching process”, with J. Bertoin and S. Mart ́ınez. Journal of Appied Probabability 45 (2008), No. 3, 714-726. 27. “A random space-time birth particle method for 2d vortex equation with external field”, with S. M ́el ́eard. 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