## tabs

**Workshop “Recent advances in mass transportation”**

The workshop will feature a series of expository mini-courses on principal directions of recent research in the broad area of mass transportation, given by renowned and active researchers in the field. These mini-courses will be accompanied by a series of talks by participants on some recent advances in particular topics related to the subject. The topics to be covered include transport, continuity and related PDEs with non smooth data, geometry of metric spaces with CD and RCD type properties, generalizations and applications of Kantorovich-Wasserstein distance in various areas of mathematics, applications of mass transportation in probability and statistics, as well as in economics, urban planning and game theory.

**The workshop is organized jointly by**

- Poncelet Center
- Higher School of Economics (Laboratory of Stochastic Algorithms and High-dimensional Inference & Mathematical Department)

**Minicourses**

- Giovanni Alberti (Università di Pisa, Italy)
- Filippo Santambrogio, (Universitè de Lyon, France)
- Dario Trevisan, (Università di Pisa, Italy)

**Speakers**

- Yurii Averboukh (IMM UB RAS)
- Bai Bo (Huawei)
- Sergey Bobkov (University of Minnesota, HSE)
- Dmitry Bukin (MSU)
- Anna Doledenok (MSU)
- Pavel Dvurechensky (WIAS)
- Thibaut le Gouic (HSE)
- Sergey Guminov (MIPT)
- Alexander Guschin (HSE)
- Nikolay Gusev (MIPT)
- Alexander Kalinin (MSU)
- Alexey Kroshnin (IITP, HSE)
- Mikhail Lifshits (SPBU)
- Mario Mariani (HSE)
- Quentin Paris (HSE)
- Fedor Petrov (St. Peterburg Department of Steklov Institute)
- Vladimir Spokoiny (WIAS, HSE)
- Mehdi Trense (ENS Paris)
- Nazarii Tupitsa (MIPT)
- Anatoly Vershik (St. Peterburg Department of Steklov Institute)
- Gershon Wolansky (Technion)
- Alexander Zimin (HSE)

To download abstracts of talks and the workshop program please follow the links at the title page of the workshop.

### Registration

### The iteration of Kantorovich-like criteria in the theory of Markov measures on the space of paths of positive graded graphs

We consider nonstationary Markov chain with finite but growing number of states. We start with some metric on the space of path and constract a sequence of metrics on that space of paths - the n-th metric is a Kantorovich metric on that space with (n-1)-th metric. The problem is to give the criteria for convergence of the sequence of metric. The theorem is a sufficient condition which is uniform compactness of iteration process.

### Coffee break

### On Random Euclidean Bipartite Matching Problems (Lecture 1)

Bipartite matching problems are widely studied combinatorial optimization problems that can be seen as special instances of optimal transport, with discrete source and target measures. Variants with random transportation costs (or measures) have been considered by many authors, with applications in statistics, e.g. by providing quantitative laws of large numbers. The random Euclidean bipartite matching problem deals with measures induced by a large number of i.i.d. uniform random variables on a domain (e.g., a cube), with natural transportation cost given by a power of the Euclidean distance.

Aim of this series of talks is to provide an overview of random Euclidean bipartite matching problems and focus on some recent challenging conjectures from [2] about quadratic transportation costs, partially proved in a joint work with L. Ambrosio and F. Stra [1].

In the first part we show how standard techniques such as sub-additivity and concentration of measure may apply (or fail), highlighting some differences with other Euclidean combinatorial optimization problems, e.g. the monopartite matching. In the second part, we describe the predictions from [1] together with their (non-rigorous) supporting arguments. In the third part, we show how to put these into a mathematical framework in the two-dimensional case.

References:

[1] L. Ambrosio, F. Stra, and D. Trevisan, A PDE approach to a 2-dimensional matching problem. Probab. Theory Relat. Fields (2019) 173: 433.

[2] S. Caracciolo, C. Lucibello, G. Parisi, and G. Sicuro, Scaling hypothesis for the Euclidean bipartite matching problem. Phys. Rev. E 90, 012118 (2014)

### Some results on the transport equation: uniqueness, mixing, and loss of regularity (Lecture 1)

Abstract. In these lectures I will describe some results and open problems on the (linear) transport equation associated to a divergence-free velocity field $u=u(t,x)$, namely $\rho_t + \nabla\cdot(\rho u) = 0$. I will focus in particular on the following issues: uniqueness, mixing, loss of regularity of the solution.These results have been obtained in collaboration with Stefano Bianchini (SISSA, Trieste), Gianluca Crippa (University of Basel), Anna Mazzuccato (Pennsylvania State University). A tentative program is the following:

**Uniqueness: **The uniqueness solution for the transport equation with given initial datum is easy to prove of the velocity field $u$ is sufficiently regular, e.g., Lipschitz in space. However, this issue turns out to be quite delicate if we consider less regular velocity fields. If the space dimension is two and $u$ is bounded and autonomous ($u=u(x)$), then it is possible to give an exact characterization of those $u$ for which uniqueness holds, and, interestingly enough, this characterization is not expressed in terms of function spaces, but by a "qualitative" property of $u$, namely a suitable weak formulation of the Sard property (by comparison, all existing results in general dimension require that $u$ belongs to some Sobolev class in space---cf. Di Perna-Lions, Ambrosio, etc.)

**Mixing: **The starting point is a conjecture by A. Bressan which states that, under certain assumptions, the "mixing scale" of the flow associated to a the velocity field $u$ decays at most exponentially in time, a statement that can be formulated in various ways in terms of the solutions of the associated transport equation. Despite the fact that this conjecture has already been proved in some relevant cases (see the work of G. Crippa and C. De Lellis) there are relatively few examples of flows which actually exhibit such exponential decay. I will illustrate some examples of this phenomenon, and in particular given by a smooth velocity field ("smooth exponential mixer").

**Loss of regularity: **Consider a solution $\rho$ of the transport equation above with initial datum $\rho_0$. If $u$ is sufficiently regular, e.g., Lipschitz in space uniformly in time, then $\rho$ is given by the composition of $\rho_0$ and the flow associated to $u$, and therefore (some) regularity of $\rho_0$ is preserved for all positive times. This result may fail completely if we $u$ belongs only to some Sobolev space that does not embed in the space of Lipschitz functions. In particular $\rho_0$ may be smooth and compactly supported, while $\rho(t,\cdot)$ does not belong to any fractional Sobolev space with order $s>0$ for any $t>0$.

### Lunch

### Improved rates for optimal matching of random samples in dimension one

We will be discussing the Kantorovich distance between random samples (drawn from a compactly supported distribution on the line) with respect to non-Euclidean metrics. Based on joint work with M. Ledoux.

### Coffee break

### Two-level duality point of view on optimal mass transportation

We discuss a theorem from abstract functional analysis which in particular case includes the Kantorovich duality theorem (the existence of optimal plan and the minimax formula for its cost). For this we introduce the space of polymorphisms (roughly speaking, the plans) and its predual, the space of so called virtually continuous functions of two variables. Based on joint works with A. M. Vershik and P. B. Zatitskii.

### On the structure of divergence-free vector measures on the plane

We will discuss the structure of divergence-free vector measures on the plane. General results in this direction were obtained by S.K. Smirnov, who proved the possibility of decomposing such measures into elementary solenoids for any finite-dimensional Euclidean space. Later further generalizations of this result were obtained by E. Paolini and E. Stepanov for acyclic Ambrosio-Kirchheim currents in Polish spaces. While in general elementary solenoids cannot be represented as closed oriented curves, we will show that in the planar Euclidean setting any divergence-free vector measure can be decomposed into such curves. The possibility of such decomposition will be closely

related to the precise characterization of the extreme points of the unit ball in BV (the space of functions of bounded variation) and, on the other hand, with the results of S. Bianchini and D. Tonon on decomposability of BV functions into a countable sum of monotone BV functions. Applications to rigidity properties of divergence-free vector measures will be discussed as well.

### Some results on the transport equation: uniqueness, mixing, and loss of regularity (Lecture 2)

### Coffee break

### On Random Euclidean Bipartite Matching Problems ( Lecture 2)

### Variational Mean Field Games from an Optimal Transport perspective (Lecture 1)

In this series of lectures I'll introduce the theory of Mean Field Games in the particular case of individual agents solving non-stochastic control problems involving the density of all the players, when the game is a potential game and the Nash equilibrium can be found by minimizing a global energy. The corresponding minimization problem is strongly related to optimal transport theory.

**Lecture 1**: Variational MFGDifferent MFG models and their variational formulations

### Lunch

### Kantorovich problems with density constraints

We discuss the Kantorovich optimal transportation problem with density constraint for measures on infinite dimensional spaces. The classical Kantorovich optimal transportation problem is a linear optimization problem on a convex domain: among all measures with fixed marginals we find an optimal one where the optimality is measured against a cost function. The modification of this problem we consider presumes a pointwise constraint on the densities of admissible measures. We have to find an optimal measure among all measures with fixed marginals and densities dominated by a given function. The talk gives a short overview of recent results in this field.

### The extreme properties of concave measures and bisection method

We consider extreme point of collections of concave measures on infinite dimensional locally convex spaces and some techniques for localization measures. We describe the most important properties for collections of measures with finite number of integral conditions. A lot of measures can be localized from whole space to smaller subspace or small convex set with the same integral conditions. There is a localization method for concave measures on infinite dimensional locally convex spaces with only one integral condition called the Bisection method. The main result is a nontrivial modification of this method in case of any finite number of integral conditions. By this algorithm we can construct the special concave measure concentrated on finite dimensional convex set and satisfied all integral conditions of original measure.

### Transportation costs for optimal and triangular transformations of Gaussian measures

The talk is devoted to connections between transportation costs (associated with the quadratic Kantorovich distance) for Monge optimal mappings and triangular mappings between Gaussian measures

### Coffee break

### Nonsmooth analysis and viability property for dynamical systems in the Wasserstein space

The concept of viability naturally appeared in the mathematical analysis of evolution. Nowadays, the viability theory has various applications in the control and game theories. The viability property states that the system starting at the given set can stay in it for some time interval. The viability property is closely connected with such concepts of nonsmooth analysis as tangent and normal cones, directional derivatives and subdifferentials due to the viability theorems which provides the equivalent formulations of viability property in the terms of nonsmooth analysis. In the talk we analyze the viability property for the mean field type control systems. They are the dynamical systems in the Wasserstein space describing the evolution of many agents in the case when the dynamics of each agent depends on her state and the distribution of all agents. Such systems can be also determined by the controlled McKan-Vlasov equation. We introduce the concepts of tangent and normal sets to the given subset of the Wasserstein space and derive the viability theorem for the mean field type control systems. Moreover, we design the feedback control assuring the viability property. The proposed control involves the extremal shift along the optimal plan.

This is joint work with Antonio Marigonda (University of Verona) and Marc Quincampoix (Universite de Brest).

### Convergence rates for empirical barycenters in geodesic spaces with lower bounded curvature

This talk presents rates of convergence for empirical barycenters over geodesic spaces with curvature lower bounds in the sense of Alexandrov. We show that fast rates of convergence are achievable under conditions related to the extendibility (in both directions) of geodesics emanating from a barycenter. Our results apply in particular to the 2-Wasserstein space where bi-extendibility of geodesics translates into regularity of Kantorovich potentials.

### Variational Mean Field Games from an Optimal Transport perspective (Lecture 2)

### Coffee break

### On Random Euclidean Bipartite Matching Problems (Lecture 3)

### On the Gilbert's mailing problem in light of the Kantorovich metric

I will review some results on limit theorems of Kantorovich metric and use them to present a convex model for optimal networks. The relations between this model and the Gilbert-Steiner problem for optimal graphs will be discussed.

### Multidimensional generalizations of Monge-Kantorovich problem

The classical Monge-Kantorovich (transportation) problem deals with measures on a product of two spaces with two independent fixed marginals. Its natural generalization (multimarginal Monge-Kantorovich problem) deals with the products of $n$ spaces $X_1, \dots, X_n$ with $n$ independent marginals. We study the Monge-Kantorovich problem on $X_1 \times X_2 \times \dots \times X_n$ with fixed projections onto the products of $X_{i_1}, \dots, X_{i_k}$ for all $k$-tuples of indices ($k < n$). We present our results for the related basic problems: feasibility, duality theorem, existence of the dual solution.

### Lunch

### Optimal Transport for Information Theory: Preliminary Thinking

Abstract: Since the original Monge formulation in 1781, the optimal transport (OT) theory has been developed to be a research area with rich results. The successful application of Wasserstein distance in generative adversarial networks (GAN) illustrates the great potential of OT in future science of information. In this report, some important problems in information theory, where some of them are open for decades, are surveyed. It is very surprising that these problems can be formulated as different OT problems. Therefore, we believe the OT formulation provides a different angle to rethink these important problems. Furthermore, these results also illustrate a possibility that the OT may lay a unified foundation for future science of information.

### Coffee break

### On the complexity of optimal transport problems

In this talk I will discuss the question of computational complexity to nd an approximation of optimal transport distance between discrete probability measures and also approximation of Wasserstein barycenter. For the optimal transport distance we analyze the entropy regularization approach combined with Sinkhorn's algorithm and accelerated gradient descent. For the barycenter problem we analyze Iterative Bregman Projection algorithm and again accelerated gradient descent.

### On Accelerated Alternating Minimization

Alternating minimization (AM) optimization algorithms have been known for a long time and are of importance in machine learning problems, among which we are mostly motivated by approximating optimal transport distances. AM algorithms assume that the decision variable is divided into several blocks and minimization in each block can be done explicitly or cheaply with high accuracy. The ubiquitous Sinkhorn’s algorithm can be seen as an alternating minimization algorithm for the dual to the entropy-regularized optimal transport problem. We introduce an accelerated alternating minimization method with a $1/k$ convergence rate, where $k$ is the iteration counter. This improves over known bound $1/k$ for general AM methods and for the Sinkhorn’s algorithm. Moreover, our algorithm converges faster than gradient-type methods in practice as it is free of the choice of the step-size and is adaptive to the local smoothness of the problem. We show that the proposed method is primal-dual, meaning that if we apply it to a dual problem, we can reconstruct the solution of the primal problem with the same convergence rate. We apply our method to the entropy regularized optimal transport problem and show experimentally, that it outperforms Sinkhorn’s algorithm.

### Lunch

### Approaches to solving OT problem

This talk deals with the comparison of Primal-Dual Accelerated Gradient Methods with Small-Dimensional Relaxation oracle recently proposed by Nesterov et al. with Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) proposed by Dvurechensky et al. Comparison is made for both entropy regularized problem and non-regularized (non-smooth) problem . The last approach takes place thanks to one of Nesterov’s methods, which is applicable to non-smooth problems. I will also consider the application of these methods to unbalanced optimal transport problem and their comparison.

### Central Limit Theorem for Wasserstein Barycenters of Gaussian Measures

Optimal transportation metrics, e.g. 2-Wasserstein metric $W_2$, 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 Frechet mean. Namely, an average of measures $\mu_1, \dots, \mu_n$ called Wasserstein barycenter is a measure which minimizes $\frac{1}{n} \sum_{i= 1}^n W_2^2(\mu_i, \nu)$ over $\nu \in P(X)$.

In stochastic setting measures are independently drawn from some distribution on $P(X)$. It is known that the Law of Large Numbers holds for Wasserstein barycenters in quite a general situation, i.e. empirical barycenter of first $n$ measures converges to the barycenter of distribution in Wassestein distance. However, the Central Limit Theorem is still an open question. This talk presents the CLT for barycenters of Gaussian measures (or, more generally, measures from a scatter-location family): we show that it holds without any additional assumptions and establish some quantitative bounds for limit distribution.

### Coffee break

### Statistical inference for barycenters (joint with Alexey Kroshnin and Alexandra Suvorikova)

Kroshnin et al (2019) established a version of a CLT for the empirical barycenters of a collection of random operators in a Bures- Wasserstein metric. In this paper we continue this study and introduce a bootstrap data-driven procedure for the constructing a data-driven condence set for the true barycenters. The talk focuses on establishing the results on bootstrap validity for this method. The analysis is heavily based on recent results on Gaussian approximation and Gaussian comparison for high dimensional random sums

### Banquet

### Some results on the transport equation: uniqueness, mixing, and loss of regularity (Lecture 3)

### Coffee break

### Variational Mean Field Games from an Optimal Transport perspective (Lecture 3)

### On constructions of stochastic processes with given terminal distribution

Many problems in the theory of stochastic processes can be treated by constructing a stochastic process from a certain class, having a given terminal distribution. For example, the famous Skorokhod Embedding Problem consists in constructing a stopped Brownian motion with a given terminal law. In this talk we study a class of single jump submartingales which have a very simple structure. We show, in particular, that the set of joint distributions of terminal values of a process and its maximum for this class coincides with the corresponding set for stopped Brownian motions

### Lunch

### Stochastic quantization and small ball probabilities

Stochastic quantization (discretization) means approximation of a random vector (often understood as a trajectory of a random process handled as an element of an infinite dimensional function space) by another random vector taking only finite number of values. The set of these values, called dictionary or codebook, may be either deterministic or constructed as a stochastic sample. In case of Gaussian vectors, the quality of quantization turns out to be tightly related to small ball probabilities, i.e. the Gaussian measures of small balls with fixed or random radii. We provide a survey of this area along with the necessary tools from the theory of Gaussian measures.

### Coffee break

### Scaling limit of a continuous model of active particles

A free energy functional arising from kinetic mean field models of interacting particles is considered. We study the variational limit, in the regime of long time and strong interaction. While the density of particles only features a weak limit in the phase space (say position and velocity), the projection of such a density on the spacial coordinates has a meaningful limit, which satisfies a hydrodynamic equation. We characterizes the tensors appearing in the hydrodynamic limit, in terms of the interaction and velocity field of the original model. Connections with transport-like metrics will be highlighted.

### Equivalent problems of barycenter in Wasserstein space and application

The barycenter of probability measure over the Wasserstein space has been introduced in a work of Agueh and Carlier (2011) for probability measure with finite support. They showed two equivalent problems: the dual and the multimarginal problem. We will present similar result for general probability measure (without finite support assumption) and discuss new application to fairness.

### Branched transportation

Branched transportation is a recent variant of optimal

transportation, inspired by the discrete Gilbert-Steiner problem. In

this model, transporting a mass $\theta$ on a distance $l$ induces a

cost $l\theta^\alpha$, where $\alpha\in (0,1)$. The sub-additivity of

this cost favours branching structures, and the transport plans will

tend to concentrate on 1-dimensional rectifiable sets. We will give an

overview on the main results available on branched transportation,

including a discussion on the landscape function, which is a kind of

potential for the branched transport energy.

**Workshop locations:**

**Cultural Center, of the Faculty of Computer Sciences of he Higher School of Economics**(Pokrovsky Boulevard, 11, building Z, 2nd floor)**Department of Mathematics of the Higher School of Economics**(Usachyova Street, 6, Moscow)