Schedule for: 22w5166 - Stochastic Mass Transports

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

"We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean field limits of recent particle filtering methods for inverse problems. We show that the transport problem splits into two separate minimisation problems: one for the evolution of mean and covariance of the interpolating curve and one its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. Also on the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed. This is joint work of Martin Burger, Matthias Erbar, Franca Hoffmann, Daniel Matthes and André Schlichting"

"It is well known that the expected Wasserstein distance between the empirical measure of $n$ i.i.d. points and the uniform measure is of order $\sqrt{\log n / n}$. However, the repulsive behavior of complex eigenvalues of random matrices forces the point process to be more evenly spread. This phenomenon will be illustrated by simulations and shall be quantified in terms of the Wasserstein distance. In this talk we investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices with i.i.d. entries and the Circular Law, the uniform distribution on the complex disk. For Gaussian entry distributions, I present an optimal rate of convergence in expected 1-Wasserstein distance of order $n^{-1/2}$, i.e. the optimal transport cost of eigenvalues is cheaper by a logarithmic factor compared to that of i.i.d. points. For non-Gaussian entry distributions with finite moments, we also show that the rate of convergence nearly attains this optimal rate."

"Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate probabilistic variant, the adapted Wasserstein distance $AW$, can play a similar role for the class $FP$ of filtered processes, i.e. stochastic processes together with a filtration. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob-decomposition, optimal stopping and stochastic control are continuous w.r.t. $AW$. We also show that $(FP, AW)$ is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace. Finally we consider computational aspects and provide a novel method based on the Sinkhorn algorithm. The talk is based on articles with Daniel Bartl, Mathias Beiglböck and Stephan Eckstein."

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

"The principal-agent problem is an important paradigm in economic theory for studying the value of private information;  the nonlinear pricing problem faced by a monopolist is a particular example.  In this lecture,  we identify structural conditions on the consumers' preferences and the monopolist's profit functions which guarantee either concavity or convexity of the monopolist's profit maximization.  Uniqueness and stability of the solution are particular consequences of this concavity.  Our conditions are similar to (but simpler than) criteria given by Trudinger and others for prescribed Jacobian equations to have smooth solutions. By allowing for different dimensions of agents and contracts,  nonlinear dependence of agent preferences on prices,  and of the monopolist's profits on agent identities,   we improve on the literature in a number of ways. The same mathematics can also be adapted to the maximization of societal welfare by a regulated monopoly. In the classical case of bilinear preferences, we introduce a new duality for certifying solutions, which leads to a free boundary formulation for the missing region in the square example of Rochet and Chone. This represents joint work with Kelvin Shuangjian Zhang."

"Lloyd S. Shapley introduced a set of axioms in 1953, now called the Shapley axioms, and showed that the axioms characterize a natural allocation among the players who are in grand coalition of a cooperative game. Recently, A. Stern and A. Tettenhorst showed that a cooperative game can be decomposed into a sum of component games, one for each player, whose value at the grand coalition coincides with the Shapley value. The component games are defined by the solutions to the naturally defined system of least squares - or Poisson - equations via the framework of the Hodge decomposition on the hypercube graph. In this talk we propose a new set of axioms which characterizes the component games. Furthermore, we realize them through an intriguing stochastic path integral driven by a canonical Markov chain. The integrals are natural representation for the expected total contribution made by the players for each coalition, and hence can be viewed as their fair share. This allows us to interpret the component game values for each coalition also as a valid measure of fair allocation among the players in the coalition. Finally, we extend the path integrals on general graphs and discover an interesting connection between stochastic integrations and Hodge theory on graphs."

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We investigate the relaxation of Optimal Transport problems when the marginal constraints are replaced by some moment constraints. Using Tchakaloff’s theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. We show the convergence of the MCOT problem toward the corresponding OT problem. In some fundamental cases, we obtain rates of convergence in O(1/n) or O(1/n^2), where n is the number of moments. We will present applications of this approach, in particular for symmetric multimarginal optimal transportation problems arising in physics. Joint work with Rafaël Coyaud, Virginie Ehrlacher and Damiano Lombardi.

Wasserstein gradient flows often arise from mean-field interactions among exchangeable particles. In many interesting applications however, the ``particles’’ are edge weights in a graph whose vertex labels are exchangeable but not the edges themselves. We investigate the question of optimization of functions over this different class of symmetries. This leads us to gradient flows or curves of maximal slopes on the metric space of continuum graphs called graphons. This is a generalization of Wasserstein calculus to higher order exchangeable structures.

"We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a finitely exchangeable sequence of $N$ random variables taking values in some Polish space $X$, we show that the law $\mu_k$ of the first $k$ components has a representation of the form $$\mu_k=\int_{\mathcal{P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda) \, \mbox{d} \alpha(\lambda)$$ for some probability measure $\alpha$ on the set of $\frac{1}{N}$-quantized probability measures on $X$ and certain universal polynomials $F_{N,k}$. The latter consist of a leading term $N^{k-1}\! /\prod_{j=1}^{k-1}(N\! -\! j) \lambda^{\otimes k}$ and a finite, exponentially decaying series of correlated corrections of order $N^{-j}$ ($j=1,...,k$). The $F_{N,k}(\lambda)$ are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals $\lambda$. Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis-Freedman between finite and infinite exchangeable laws."

"The existence of a transport map from the standard Gaussian leads to succinct representations for, potentially complicated, measures. Inspired by results from optimal transport, we introduce the Brownian transport map that pushes forward the Wiener measure to a target measure in a finite-dimensional Euclidean space. Using tools from Ito's and Malliavin's calculus, we show that the map is Lipschitz when the target measure satisfies an appropriate convexity assumption. This facilitates the proof of several new functional inequalities. In other settings, where a globally Lipschitz transport map cannot exist, we derive Sobolev estimates which are intimately connected to several famous open problems in convex geometry.  Joint work with Yair Shenfeld"

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The existence of Lipschitz transport maps between probability measures leads to transportation of functional inequalities from the source to the target measure. Finding such maps, however, is a non-trivial problem. One successful example is Caffarelli's contraction theorem: The optimal transport map is 1-Lipschitz if the source measure is Gaussian and the target measure is strongly log-concave. On the other hand, if this stringent assumption on the target measure is omitted, much less is known about the Lipschitz properties of the optimal transport map. We show, together with Dan Mikulincer, that these questions can be resolved if we work with transportation of measure along Langevin dynamics rather than with optimal transport. In particular, we show that the Langevin dynamics yields Lipschitz transport maps from the Gaussian to target measures which are either semi-log-concave with bounded support, or mixtures of Gaussians. In turn, this transport map leads to improved functional inequalities. Time permitting, I will talk about further work in progress with Max Fathi, Dan Mikulincer, and Joe Neeman.

We discuss an optimal Brownian stopping problem from a given initial distribution where the target distribution is free and is conditioned to satisfy a given density height constraint. The solutions to this optimization problem then generate solutions to the Stefan problem, a free boundary problem of the heat equation that describes supercooled fluid freezing (St1) or ice melting (St2), depending on the type of cost for optimality. The freezing (St1) case has not been well understood in the literature beyond one dimension, while our result gives a well-posedness of weak solution in general dimensions, with naturally chosen initial data. We also give a new connection between the freezing and melting Stefan problems. This is joint work with Inwon Kim (UCLA).

"We deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modelled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE. In standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. The results are illustrated with Wasserstein perturbations of Lévy processes, Ornstein-Uhlenbeck processes, and Koopman semigroups. The talk is based on joint works with Daniel Bartl, Stephan Eckstein, Sven Fuhrmann, and Max Nendel."

In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. For $\mu$, $\nu$ probability measures on $\mathbb{R}^d$, increasing in convex order, stretched Brownian motion [1] provides an analogue for the martingale version of this problem. In dimension $d=1$ it was shown in [2] that any martingale transport decomposes into at most countably many invariant intervals and that this decomposition is universal. Extensions of this result to $d \geq 2$ were studied in [4], [5], [3]. We show that the dual optimization problem attached to a stretched Brownian motion induces the universal DeMarch--Touzi paving [3] of $\mathbb{R}^d$. Joint work with M. Beiglböck, J. Backhoff, and B. Tschiderer References: [1] J. Backhoff-Veraguas, M. Beiglböck, M. Huesmann, and S. Källblad. Martingale Benamou-Brenier: a probabilistic perspective. Ann. Probab., 48(5):2258–2289, 2020. [2] M. Beiglböck and N. Juillet. On a problem of optimal transport under marginal martingale constraints. Ann.Probab., 44(1):42–106, 2016. [3] H. De March and N. Touzi. Irreducible convex paving for decomposition of multidimensional martingale transport plans. Ann. Probab., 47(3):1726–1774, 2019. [4] N. Ghoussoub, Y.-H. Kim, and T. Lim. Structure of optimal martingale transport plans in general dimensions. Ann. Probab., 47(1):109–164, 2019. [5] J. Obłój and P. Siorpaes. Structure of martingale transports in finite dimensions. arXiv:1702.08433, 2017.

"We introduce a framework that allows to employ (non-negative) measure-valued processes for energy market modeling, in particular for electricity and gas futures. Interpreting the process’ spatial structure as time to maturity, we show how the Heath-Jarrow-Morton approach (see Heath et al. (1992)) can be translated to this framework, thus guaranteeing arbitrage free modeling in infinite dimensions. We derive an analog to the HJM-drift condition and then treat in a Markovian setting existence of (non-negative) measure-valued diffusions that satisfy this condition. To analyze mathematically convenient classes we consider measure-valued polynomial and affine diffusions where we can precisely specify the diffusion part. ndeed, it depends on continuous functions satisfying certain admissibility conditions. For calibration purposes these functions can then be parametrized by neural networks yielding measure-value analogs of neural SPDEs. Exploiting on the one the analytic tractability coming from the affine and polynomial nature and on the other hand stochastic gradient descent methods for neural networks allows for efficient mass transport in infinite dimensions from the initial distribution of the measure-valued process encoding the current forward curve to the distribution of future forward curves implied from option prices. The talk is based on joint work with Luca Di Persio and Francesco Guida"

(Joint work with Martin Brückerhoff and Martin Huesmann.) I will present a systematic construction for families of real random variables $(X_t)_{t \geq 0}$ that 1) are a martingale 2) go through any given family of marginals $(\mu_t)_{t \geq 0}$ —provided the necessary condition that $\mu_t$ is increasing (for $t$) in convex order. Our construction method is based on the shadow martingale couplings introduced in a previous work with Mathias Beiglböck. We show optimality properties and a Choquet representation for the constructed martingale.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

"We introduce a stochastic control problem for measure-valued martingales, which are stochastic processes taking values in the space of probability measures and satisfying a martingale-like property. Such control problems arise naturally in a variety of applications, such as the Skorokhod Embedding problem, and informational games with asymmetric information. Our main results concern characterisation of the value of the problem in terms of the viscosity solution to a corresponding PDE. Joint work with Sigrid Källblad, Martin Larsson and Sara Svaluto-Ferro."

In this talk we consider a class of probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We will discuss various structural properties of such processes; in particular, we present an appropriate version of Itô’s formula for controlled MVMs and an existence result for an SDE featuring MVMs. We also illustrate how problems of this type arise in a number of applications paying particular attention to their connection to some functional inequalities. The talk is based on joint work with Alex Cox, Martin Larsson and Sara Svaluto-Ferro.

Given any deep fully connected neural network, initialized with random Gaussian parameters, we bound from above the quadratic Wasserstein distance between its output distribution and a suitable Gaussian process. Our explicit inequalities indicate how the hidden and output layers sizes affect the Gaussian behaviour of the network and quantitatively recover the distributional convergence results in the wide limit, i.e., if all the hidden layers sizes become large. Joint with A. Basteri (U. Pisa)

The optimal matching problem is a classical random combinatorial problem which may be interpreted as an optimal transport problem between random measures. Recent years have seen a renewed interest for this problem thanks to the PDE ansatz proposed in the physics literature by Caracciolo and al. and partially rigorously justified by Ambrosio-Stra-Trevisan. In this talk I will show how this ansatz combined with subadditivity may be used to give information both on the optimal cost and on the structure of the optimal transport map at various scales. This is based on joint works with L. Ambrosio, M. Huesmann, F. Otto and D. Trevisan.

"The optimal matching problem is a classical random variational problem that received interest in the last 30 years. We show that there exists no cyclically monotone invariant matching of two independent Poisson processes in the critical dimension $d=2$. Our argument relies on a recent harmonic approximation theorem together with the two-dimensional local asymptotics for the bipartite matching problem, for which we provide a new self-contained proof based on martingale arguments. Joint work with M. Huesmann (WWU Münster) and F. Otto (MPI Leipzig)"

This talk discusses new quantitative results on mean field limits for large $n$-particle diffusion systems. The "distance" between the marginal law of $k$ particles and its limiting product measure is shown to be $O((k/n)^2)$ at each time, as long as the same is true at time zero. A simple Gaussian example shows that this rate is optimal. The best previously known bound was $O(k/n)$ and was widely believed to be optimal. Various "distances" are possible here, with relative entropy playing a key role: The new approach relies on establishing transport inequalities for the limiting measure, as well as a differential inequality which bounds the $k$-particle entropy in terms of the $(k+1)$-particle entropy, derived from a form of the BBGKY hierarchy.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this work, we look at the problem of robust pricing and hedging American-style options whose underlyings are modelled by continuous semimartingales. Similar to previous related works in discrete time (e.g., Aksamit et al (2019)), we enlarge the probability space with the stopping decision and use path-dependent optimal transport (Guo and Loeper (2021)) to establish pricing-hedging dualities for European claims in the enlarged space. To connect this to the original space, we show that martingale measures in the enlarged space can be expressed as convex combinations of martingale measures in the original space coupled with true stopping times. This combination is identified by decomposing the Azéma supermartingale of the associated random time. This in turn allows us to prove the pricing-hedging duality for American options in continuous time.

Let π ∈ Π(μ, ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

"Given a vector $\mu = (\mu_i)_{i=1}^n$ of probabilities, consider an optimization problem of the form $$\mathcal{P}:=\mathrm{inf} \mathbb{E}[c(X)],\qquad (\mathrm{P})$$ where the infimum runs over all the (laws $\pi$ of) random vectors $X = (X_i)_{i=1}^n$ with marginals $X_i \sims \mu_i$, and which satisfy some additional linear constraints. When $\mu$ is finitely supported, (P) is a linear program, and thus is well understood and can be solved numerically with high efficiency. It is then of interest to approximate general $\mu$ with some finitely supported $\mu^k$ which satisfy the same linear constraints, ideally in a way that the optimal values $\mathcal{P}(\mu^k)\to\mathcal{P}(\mu)$ converge as $k\to\infty$ and so do the corresponding minimisers $\pi^k\to\pi$, and possibly $\mu^k$ satisfy some optimality property and/or some additional constraint. We consider the case where $X$ is required to be a vector-valued martingale, thus constructing a quantization method which preserves the convex order on measures on a separable Banach space. This is joint work with Marco Massa."

We discuss entropically regularized optimal transport and its stability with respect to the marginals. A qualitative result (for weak convergence) is obtained using the geometric notion of c-cyclical monotonicity and a quantitative result (for Wasserstein distance) is obtained by control theoretic methods. These results can be applied to deduce convergence of Sinkhorn’s algorithm for unbounded cost functions such as the quadratic cost and find a convergence rate in Wasserstein sense. Based on joint works with Espen Bernton, Stephan Eckstein, Promit Ghosal, Johannes Wiesel.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

The aim of this talk is to present two set of results on the Schrödinger bridge problem that can be obtained leveraging logarithmic Sobolev inequalities, both in their integrated and local form. The first result is about the short time (small noise) convergence of the gradient of Schrödinger potentials to the Brenier map. The second set of results is about the stability of Schrödinger bridges and the entropic cost with respect to perturbations in the marginal measures. Based on a joint work with A.Chiarini,G.Greco and L.Tamanini.

When approximating in Wasserstein distance the two marginals of a martingale coupling by probability measures in the convex order, it is possible to construct a sequence of martingale couplings between these probability measures converging in adapted Wasserstein distance to the original coupling. We deduce the stability with respect to the marginal distributions of the Weak Martingale Optimal Transport problem in dimension one.

Localisation is a powerful tool in proving and analysing various geometric inequalities, including isoperimertic inequality in the context of metric measure spaces. Its multi-dimensional generalisation is naturally linked to optimal transport of vector measures and vector-valued Lipschitz maps. I shall present recent developments in this area: a partial affirmative answer to a conjecture of Klartag concerning partitions associated to Lipschitz maps on Euclidean space, and a negative answer to another conjecture of his concerning mass-balance condition for absolutely continuous vector measures. During the course of the talk I shall also discuss an intriguing notion of ghost subspaces related to the above mentioned partitions.