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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 BackhoffVeraguas 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 realworld 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 (20142015). 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 UChileCNRS. University of Chile.  20032011: Assistant Professor, Department of Mathematical Engineering, University of Chile. I. EDUCATION: • Graduate Studies: Ph.D. in Mathematics, Universit ́e de Paris 6, 19992004. • Undergraduate studies: Mathematical Engineering. Universidad de Chile, 19931999. 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 continuousstate branching process”, with D. Fekete and A. E. Kyprianou. arxiv.org/abs/1702.03533. Submitted 3. “RayKnight representation of flows of branching processes with competition by prun ing of L ́evy trees” with J.Berestycki and M.C. Fittipaldi. arXiv:1506.00046 . In revision for Probability Theory and Related Fields. 4. “Quantitative uniform propagation of chaos for Maxwell molecules”, with R.Cortez. arXiv:1512.09308. Accepted with minor corrections in Communications in Mathe matical Physics. 5. “Spectrotemporal Power Signature Estimation on SeismicCoda Data”, (2nd author) with G.Soto (first author), and S.Gaete. Accepted in 9th International Symposioum on Rockburst and Seismicity in Mining RASIM, Santiago, Nov. 2017. 6. “A Seismic Datadriven Methodology to Assess the Probability of Softening and Hard ening Regimes”, (2nd author) with G.Soto (first author), S.Gaete , J.Prado and R.Dunlop. Accepted in 9th International Symposioum on Rockburst and Seismicity in Mining RASIM, Santiago, Nov. 2017. 7. “A non disruptive reliability approach to assess the health of microseismic sensing networks”, with D. Neira, G. Soto, J. Prado and S. Gaete. To appear in Applied Stochastic Models in Business and Industry 8. “A variational approach to some transport inequalities” with N.Gozlan and J.F. Jabir. To appear in Annals de l’ Institut Henry Poincar ́e Probab. Stat.. 9. “Quantitative exponential bounds for the renewal theorem with spreadout distributions”, with J.B. Bardet and A.Christen. Markov Process. Relat. Fields 23 (2017), 67–86. 10. “Rate of convergence to equilibrium of fractional driven stochastic differential equa tions with some multiplicative noise” with F.Panloup. Annals de l’ Institut Henry Poincar ́e Probab. Stat.. 53 (2017), no. 2, 505 – 538 11. “Robust utility maximization without model compactness ”, with J.Backhoff. SIAM J. Financial Math. 7 (2016), no. 1, 70–103. 12. “Long time behavior of telegraph processes under convex potential” , with H. Gu ́erin and F. Malrieu. Stochastic Process. Appl. 126 (2016), no. 10, 3077–3101. 13. “A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations”, with B.Jourdain. Ann. Probab. 44 (2016), no. 1, 131–170. 14. “Quantitative propagation of chaos for generalized Kac particle systems ”with R. Cortez. Ann. Appl. Probab. 26 (2016), no. 2, 892–916. 15. “An online twostage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mine”, with G. Soto, R. Cortez and L. Mujica, Measurement. 92 (2016), pag. 340–351. 16. “Nonlocal LotkaVolterra systems with cross diffusion in heterogeneous medium”, with S. Meleard. J. Math. Biol. 70 (2015), no. 4, 829–854. 17. “Long time behavior of stochastic vortex systems”, with B.Jourdain. Markov Process. Related Fields 20 (2014), no. 4, 675  704. 18. “Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain”, with M.Bossy, P.E. Jabin and J.F. Jabir. Communica tions in Partial Differential Equations Vol. 38, 7, (2013) 11411182. 19. “Quantitative estimates for the long time behavior of an ergodic variant of the tele graph process”, with H. Gu ́erin and F. Malrieu. Advances in Applied Probability Vol. 44, No. 4 (2012), 977994. 20. “On SDE associated with continuousstate branching processes conditioned to never be extinct”, with M.C. Fittipaldi. Electron. Commun. Probab. 17 (2012), No. 49, 113. 21. “Stochastic vortex method for forced three dimensional NavierStokes equations and pathwise convergence rate”, Annals of Applied Probability, Vol. 20, (2010), No. 5, 1761 1800. 22. “Energy efficiency of consecutive fragmentation processes”, with N. Krell, and Se. Mart ́ınez. Journal of Applied Probability, 47 (2010), No. 2, 543561. 23. “The limiting movetofront searchcost in law of large numbers asymptotic regimes”, with J.Barrera. Annals of Applied Probability 20, (2010), No. 2, 722755. 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) 124133. 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. 34, (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, 714726. 27. “A random spacetime birth particle method for 2d vortex equation with external field”, with S. M ́el ́eard. Mathematics of Computation 77 (2008), No. 263, 15251558 . 28. “Paths clustering and an existence result for stochastic vortex systems”, with M. Mart ́ınez, Journal of Statistical Physics 128 (2007), No. 3, 699719,. 29. “Probabilistic interpretation and stochastic particle approximations of three dimen sional NavierStokes equations”, Probability Theory and Related Fields 136 (2006), No.1, 102156. 30. “Uniqueness for a weak non linear evolution equation and large deviations for diffusing particles with electrostatic repulsion”, Stochastic Processes and their Applications, 112 (2004) , No.1, 119144. 31. “The homotopical reduction of a nearest neighbor random walk”, with Se. Mart ́ınez, Bulletin of the Brazilian Mathematical Society, New Series 34, (2003), No.3, 509528. 32. “Nonlinear martingale problems involving singular integrals”, Journal of Functional Analysis 200, (2003) No.1, 198236.
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