Plenary Conferences



Franck Barthe

Title: “Gaussian kernels also have Gaussian minimizers”
The title hints at E. Lieb’s celebrated result on maximizers of multilinear gaussian kernels acting on products of L_p spaces. In a joint work with P. Wolff, we put forward a similar principle for minimizers when the indices p are less than 1 (possibly negative). This extends and unifies many “reverse” inequalities known in the literature, and pertaining to analysis and probability theory. Monotone transportation maps play a key role in the proof, and the presence of negative coefficients requires new ingredients.


Dominic Schuhmacher

Title: “Bounding Wasserstein metrics for spatial point processes”
Stein’s method (Stein, 1972) is a versatile technique for computing explicit upper bounds on distances between an “involved” and a more tractable probability distribution. From early times it has been particularly successful for metrics of L_1-Wasserstein type (Erickson, 1974). In the present talk I give a review of Stein’s method for distributions on quite general metric spaces, with a strong emphasis on point process distributions, which has its origins in Barbour and Brown (1992). We discuss a choice of metric between point patterns that is reasonable both from a probabilistic and an applied point of view. In addition to various established bounds on Wasserstein distances between point process distributions, I also present a very recent result about Poisson process approximation of point processes that have been randomly thinned by a [0,1]-transformed Gaussian random field.



Sanvesh Srivastava 

Title: “A Divide-and-Conquer Bayesian Approach to Large-Scale Kriging
Flexible hierarchical Bayesian modeling of massive data is challenging due to poorly scaling computations in large sample size settings. We are motivated by Bayesian spatial process models for analyzing geostatistical data, which are typically posed as nonparametric regression problems with a Gaussian process prior on the unknown function. Posterior computations in such models become prohibitive as the number of spatial locations becomes large. We propose a three-step divide-and-conquer strategy within the Bayesian paradigm to achieve massive scalability for any nonparametric regression model based on Gaussian process priors. We partition the data into a large number of subsets, apply a readily available Bayesian spatial process model on every subset in parallel, and combine the posterior distributions estimated across all the subsets via the Wasserstein barycenter. The barycentric posterior distribution conditions on the entire data and is used for prediction and inference. We call this approach “Distributed Kriging” (DISK), and it offers significant advantages in applications where the entire data are or can be stored on multiple machines. Under the standard theoretical setup, we show that if the number of subsets is not too large, then the Bayes risk of estimating the true spatial surface using the DISK posterior distribution decays to zero at a nearly optimal rate. We also demonstrate its empirical performance using stationary and non-stationary Gaussian process priors. A variety of simulations and a geostatistical analysis of the Pacific Ocean sea surface temperature data validate our theoretical results.
This presentation is based on a joint work with Raj Guhaniyogi (University of California Santa Cruz),  Cheng Li (National University of Singapore), and Terrance Savitsky (US Bureau of Labor Statistics). A preliminary version of the manuscript is available on Arxiv at


Jean Michel Loubes

Title: “Law of Large Number for Empirical Wasserstein’s Barycenter and Applications in Statistics”
(Joint work with T. Le Gouic)
Based on the Fréchet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random probabilities defined on a geodesic space (E, d). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the probability measures or of a growing set of probability measures. We use this property for some statistical applications : Gaussian processes indexed on distributions and inverse problems for distributions.


Johan Segers 

Title: “Tail behaviour of a multivariate quantile based on optimal transports”
(Joint work with Cees de Valk (KNMI) )
Chernozhukov et al. (Annals of Statistics, 2017) proposed the notion of Monge-Kantorovich quantile contours of a multivariate distribution. We propose a variation of their definition that is better suited to heavy-tailed distributions since it does not require moment constraints. The reference measure we consider is different from theirs as well. We show that our definition is stable with respect to M_0 convergence of measures (Hult and Lindskog, Publ. Inst. Math. (Beograd), 2005), a mode of convergence that allows for infinite limit measures and which underpins the theory of multivariate regular variation. Moreover, the optimal transport plan of a multivariate regularly varying measure is shown to have at most linear growth. In combination, we find that tail quantile contours of a multivariate regularly varying distribution converge to a limiting shape as the tail probability tends to zero. We conclude with some ideas about nonparametric estimation of the tail quantile contours.


Eustasio del Barrio

Title: “Central Limit Theorem for empirical transportation cost in general dimension”
(Joint work with J.M. Loubes)
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on \($\mathbb{R}^d$, with $d\geq 1$\). We provide new results on the uniqueness and stability of the associated optimal transportation potentials, namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.


Juan A. Cuesta Albertos

Title: “Clustering probabilities: a big data perspective”
(Joint work with E. del Barrio, C. Matrán, and A. Mayo Íscar)
Let us assume that we have a big data set obtained from a mixture of k probability distributions, and that in order to ease its handling, it was divided in m manageable subsets, which were analyzed by separate units. Consequently, each of the m computational units  provided, as a result of its analysis, k different probability distributions.  The problem is to find a mixture of k distributions to summarize the solutions provided by the units.
We propose a kind of trimmed k-means clustering procedure, which includes the possibility to delete some of the most discrepant  distributions sent by the units.
Obviously, this approach enables us to deal with  units  sending wrong components of the mixture, but, it also allows us to handle units which made a wrong selection of k.
The procedure  results in a gain in stability and robustness, highly convenient in this setting. We illustrate the methodology with simulated and real data examples.


Quentin Paris

Title: “On the rate of convergence of empirical barycenters”
(Joint work with Adil Ahidar and Thibaut Le Gouic)
In this talk, we investigate variance inequalities in spaces with curvature bounded from above by a positive constant. In particular, we show that an upper bound on the curvature of a metric (and measured) space implies a variance inequality which further allows to derive rates of convergence for empirical barycenters. Our results find a natural application by providing non-asymptotic rates of convergence of empirical barycenters in Wasserstein spaces. We also connect curvature bounds on the Wasserstein space with commutativity of the optimal transport plans between measures.


Alexandra Suvorikova

Title: “CLT for barycenters in 2-Wasserstein space”
Optimal transportation metrics is a natural way to measure distance between probability distributions on some space X taking into account the underlying geometry of X. Moreover, it gives a possibility to define a nonlinear, “geometrical” averaging of measures via Fréchet mean.In the present work we consider a stochastic setting on 2-Wasserstein space. It is known that the Law of Large Numbers holds for Wasserstein barycenters in quite a general case, i.e. an empirical barycenter of n measures converges to the population barycenter in Wassestein distance. However, the Central Limit Theorem is still an open question. The work presents CLT for barycenters of measures sampled from a location-scatter family: we show that it holds without any additional assumptions and establish some quantitative bounds for a limiting distribution.