Optimal Transportation and Applications

Contribution List

22. Approximation methods in optimal transport and applications to Sobolev calculus and stability

Francesco Nobili

A classical characterization of Sobolev functions relies on absolute continuity along almost every line parallel to the coordinate axes. Over the past decades, this viewpoint has inspired extensive research in the analysis of metric spaces, revealing deep connections between optimal transport, Sobolev calculus, and Ricci curvature.
After a brief review of these ideas, I will present an...

20. Conservation laws on the Wasserstein space: from selection principles in Mean Field Games towards entropy solutions

Alpar Meszaros

As argued by Harsányi and Selten, selecting one Nash equilibrium from the set of all equilibria is one of the most fundamental questions in game theory. Introduced around two decades ago, Mean Fields Games (MFGs) aim to model symmetric large population dynamic games, where the agents are interacting via the distribution of their states and controls. While there is a very good theory for MFGs...

8. Convex relaxation of the Arnold-Moffatt system

Yann Brenier

The Arnold-Moffat system can be interpreted as a gradient flow for divergence-free vector
fields that dissipates their kinetic energy while preserving their topology leading in infinite time to
a stationary solution of the Euler equations of prescribed topology.
In our talk, we first place the Moffatt equations inside a larger framework of “incompressible wave maps” and derive them...

2. Decomposition of measures by their geometric tangent cone

Averil Aussedat

The geometric tangent cone to the Wasserstein space is known to be isometric to a set of plans since the work of Gigli (2008). It is a nonlocal object, in that restricting the measure on a subset might change its tangent cone substantially. However we show that the subset of centred elements, those with barycenter 0 in each fiber, is stable by restriction. Following this we compute a...

7. Entropic Lorentz optimal transport and the Sinkhorn algorithm

Mathias Braun

We study an entropic regularization of optimal transport on spacetimes and directed acyclic graphs, where mass may only be transported along a causal relation, the cost is a power of the time separation, and one maximizes rather than minimizes the total cost (or benefit). Unlike classical entropic optimal transport, the causal constraint makes the cost take the value minus infinity, so...

28. Finite speed Ornstein-Uhlenbeck-Kac processes and their applications in generative modeling

Gabriele Steidl

In score-based diffusion models, the underlying probability path is governed by an Ornstein-Uhlenbeck (OU) process,
connecting a target data variable with a Gaussian.
This variance-preserving OU path is driven by the Wiener process,
and it is known that due to its unbounded variation,
the OU path may admit certain singularities concerning its score at times close to the data.
To...

35. First order calculus on low regularity Riemannian manifolds

Vanessa Ryborz (Universität Wien, Austria)

The first order calculus on metric measure spaces by Ambrosio-Gigli-Savaré, after Cheeger turned out to be a powerful tool, for instance in the presence of curvature-dimension conditions. I study the first order calculus on smooth manifolds endowed with Riemannian metrics of low regularity and compare it to the classical first-order calculus provided by the differential structure of the...

23. From Brenier on iterated Wasserstein spaces to lightspeed calibration of stochastic processes

Gudmund Pammer

We establish a Brenier theorem and a superposition principle for iterated Wasserstein spaces. Specifically, for a separable Hilbert space $H$ and $N\ge 1$, we construct a full-support probability $\Lambda \in P_2^{N}(H)= P_2(\ldots P_2(H)\ldots)$ that is transport regular: for all $P,Q\in P_2^{N}(H)$ with $P\ll \Lambda$, the $W_2^2$-optimal transport from $P$ to $Q$ is unique and of Monge...

18. Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions

Anna Korba

The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full...

24. Inverse optimal transport

Clarice Poon

In this talk, I will discuss a particular inverse problem arising from optimal transport (OT). In many modern pipelines, however, the transport cost is not known a priori and must be inferred from data. This leads to inverse optimal transport (iOT): recover the ground cost (or metric parameters) from an observed optimal coupling. On the other hand, modern computational pipelines typically...

32. Lambda-geodesic convexity of the squared sliced Wasserstein distance

Giacomo Cozzi (Università di Padova)

The sliced Wasserstein distance was introduced as an analogue of the Wasserstein distance, based solely on one-dimensional transports. This makes it computationally efficient and well suited to applications.
We prove that the squared sliced Wasserstein distance from a fixed target measure is (\lambda)-geodesically convex, provided that the target satisfies a geometric condition controlling...

14. Li-Yau-Hamilton and Aronson-Bénilan estimates in the JKO scheme

Fanch Coudreuse

The Li–Yau and Aronson–Bénilan estimates provide explicit lower bounds on the Laplacian of the pressure associated with solutions to the heat and porous medium equations. A key feature of these estimates is their independence from the initial datum. They can therefore be used to establish various regularization effects, quantitative Harnack inequalities, and long-time asymptotics for these...

11. Localisation for the second-order Beckmann problem: bimartingale couplings and leaf decompositions

Krzysztof Ciosmak

Classical localisation in optimal transport decomposes transport into monotone couplings along transport rays. In this talk, I will present a second-order analogue: transport rays are replaced by the leaves of the 1-Lipschitz gradient Du of an optimal dual potential u in C^(1,1)(R^n), while monotone couplings are replaced by bimartingale couplings. This framework applies to the three-marginal...

25. Long-time behaviour and asymptotic geometry of nonlocal aggregation equations with saturating mobility

Emanuela Radici

Nonlocal aggregation equations with nonlinear mobility arise naturally
in the study of interacting particle systems subject to congestion and density
constraints. Their evolution can be formally interpreted as a gradient
flow of the interaction energy with respect to a generalized transport geometry
induced by the mobility.
In this talk, I will investigate the long-time behaviour of...

10. Near-Lipschitz stability of the Kim–Milman map

Sinho Chewi

The Kim–Milman construction provides a transport map T_μ from a reference, taken here to be standard Gaussian, to a target measure μ over R^d. We show that under the same condition in the literature which implies that T_μ is Lipschitz, when μ is perturbed to another measure ν, then the L^2 difference between T_μ and T_ν is controlled by W log(1 + 1/W) where W is the Wasserstein distance...

9. Nonlinear kinetic Fokker-Planck equations as gradient flows of the free energy

Guillaume Carlier

In this talk, I will consider nonhomogeneous kinetic equations that involve a free transport operator and a diffusion of porous medium type acting on velocities. The main novelty is a gradient flow interpretation of dynamics driven by an interplay of conservative and dissipative effects. Relying on a notion of discrepancy adapted to a phase space of positions and velocities, built upon...

1. On quantum optimal transport and beyond

Nataliia Monina

In recent years, convex regularization has gained a lot of attention in the context of optimal transport (OT). In this talk, we shall see how these ideas have transcended from the classical theory into the world of quantum optimal transport (QOT). Furthermore, we will look beyond QOT and consider a broader concept of optimization problems known as semidefinite programming (SDP). We begin by...

4. On the geometries of Wasserstein spaces

Charles Bertucci

I will distinguish two geometries on Wasserstein spaces, one coming from the linear structure of signed measures, and one coming from the geometry of the base space. I will explain why a Radon-Nikodym property holds in both geometries, and draw their usual consequences. I will then spend some time explaining what I believe are natural expression of two "geometric" objects in this second...

33. On the stability of local Riemannian Ricci curvature lower bounds

Federico Renzi (Scuola Normale Superiore)

The RCD condition provides a notion of Riemannian-like metric measure space satisfying synthetic Ricci curvature lower bounds. In this talk I will present a notion of RCD condition localized on an open subset of the ambient space, with variable curvature lower bound, by requiring infinitesimal Hilbertianity and displacement convexity of suitable entropy functionals on neighbourhoods of every...

21. Optimal transport on null hypersurfaces and the null energy condition

Andrea Mondino

The talk introduces a framework for studying the geometry of lightlike (null) hypersurfaces in spacetimes that may be singular or lack a smooth structure. Building on tools from optimal transport and modern Lorentzian geometry, we develop a flexible, synthetic approach to describing such hypersurfaces without relying on the usual smoothness assumptions.
Within this framework, we propose a...

26. Properties perimeter (quasi)minimizers in metric spaces

Tapio Rajala

We will revisit the regularity properties of (quasi)minimizers and consider two types of minimizers involving the perimeter. One minimization is for the perimeter among sets with fixed volume, leading to isoperimetric sets and the other is a minimization of the perimeter with an additive term penalizing the deviation in measure from a given set.

In essentially non-branching MCP(K,N)-spaces,...

16. Regularity and structure of spaces with synthetic Ricci bounds and positive injectivity radius

Shouhei Honda

In this talk, we introduce a structure theory of metric measure spaces with Ricci curvature bounded below and positive injectivity radii. In particular, we show that such spaces are weighted Riemannian manifolds of W^{1,p}-regularities for any finite p. Applications include a resolution to a conjecture of V. Kapovitch via a metric smoothing. This is a joint work with Ruobing Zhang (US San...

29. Sectional Curvature for Kantorovich-Wasserstein and Hellinger-Kantorovich Geometries

Karl-Theodor Sturm

We derive an explicit formula for the sectional curvature of the space M_0(X) of finite measures on a Riemannian manifold X. The space M_0(X) is equipped with the Hellinger-Kantorovich metric HK. Even in the case X = Rn, the curvature is comprised of two parts: the ‘lifted part’ is negative, and the ‘twisted part’ is positive. It will be analyzed in detail for the multidimensional torus.
Our...

12. Stability of entropic projections

Giovanni Conforti

Entropy minimization problems under marginal constraints play a central role in machine learning, data science, and computational optimal transport. Prominent examples include entropic optimal transport and the (regularized) Wasserstein barycenter problem. This talk presents general principles for establishing quantitative stability of entropic projections. In particular, I will show how...

17. TBA

Martin Huesmann

30. TBA

Maxime Sylvestre

31. TBA

Dario Trevisan

34. TBA

Prof. Gero Friesecke (Technische Universität München)

6. The Monge -Kantorovich tangent bundle to a measure

Guy Bouchitté

Inspired by Monge-Kantorovich optimal transport theory, we present a notion of tangency to a measure based on the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures with respect to the $W_1$-distance. For any Radon measure $\mu$ in $Rd$, a tangent bundle $T_\mu$ can be defined...

19. The monopolist's free boundary problem in the plane: an excursion into the economic value of private information

Robert McCann

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 one example; others include optimal taxation and auction design.
For multidimensional spaces of consumers (=agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of...