Optimal transport has a broad scope of applications from biology, finance, to machine learning/AI, This workshop will bring researchers at the frontier of computational optimal transport. Major directions to be considered include stability of the optimal solutions, numerical methods for stochastic optimal transport, and geometry of the space of stochastic processes. Participants in our workshop will interact with those in “Foundations of Data Science and Machine Learning”, “Information-Based Complexity”, “Stochastic Computation”, “Continuous Optimization”, “Computational Dynamics”, “Inverse Problems”, among others.
Organizers
Speakers
Semi-plenary speakers
University of Vienna
University Washington
Invited speakers
University of Vienna
University of Padova
CALTECH
INRIA Saclay
INRIA
Columbia University
University of Bonn
Université Paris-Saclay
University of Vienna
Inria
TU Graz
University of Alberta
University of Vienna
TU Wien
Carnegie Mellon University
Università Cattolica del Sacro Cuore
Thursday, 9 July
![[HS 13 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS13.png){kind=link}
14:00-14:30
Walter Schachermayer (University of Vienna)
Given µ and ν, probability measures on Rd in convex order, a Bass martingale is arguably the most natural martingale starting with law µ and finishing with law ν. Under a (necessary and sufficient) irreducibility condition on (µ,ν) there exists a unique Bass martingale. It depends crucially on its Bass measure α, the starting law of the accompanying Brownian motion. We identify a functional on the Wasserstein space of probability measures on Rd which attains its minimum at the Bass measure α. We also show that there is an induced corresponding flow converging to α.
Joint work with J. Backhoff, M. Beiglböck, G. Pammer, B. Tschiderer.
14:30-15:00
Katharina Eichinger (INRIA Saclay)
Finding regular transport maps between measures is an important task in generative modelling and a useful tool to transfer functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that for our purposes optimality of the transport map does not play a role. This is why several works investigate other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we establish a lower bound on the log-semiconcavity along the heat flow for a class of what we call asymptotically log-concave measures. We will see that this implies Lipschitz bounds for the heat flow map introduced by Kim and Milman. We will also show that these log-semiconcavity bounds are sufficient for stability of these maps in entropy and Wasserstein distance. Based on a joint work with Louis-Pierre Chaintron and Giovanni Conforti, and a joint work with Sinho Chewi and Aram-Alexandre Pooladian.
15:00-16:00 semi-plenary talk
Soumik Pal (University of Washington)
Suppose we are interested in sampling from a multidimensional density. The most common method is to consider some time discretization of the Langevin diffusion, such as constructing a suitable Markov chain approximation. We consider the possibility of deforming the underlying geometry of the Euclidean space and consider instead sampling using a particular Markov chain approximation of the so-called Mirror Langevin diffusions, first introduced in the context of machine learning in 2020/21. I will give an overview of this family and how it relates to Wasserstein gradient flows, Schrödinger bridges, Sinkhorn algorithms, and many other topics of recent interest.
16:00-16:30 Coffee Break
16:30-17:00
Alberto Gonzalez Sanz (Columbia University)
Optimal transport (OT) is a central tool in data science. A common approach is to regularize OT with an entropy. The most widely used choice is the KL entropy, which leads to entropic optimal transport (EOT). From a practical standpoint, EOT is successful for two main reasons: the dual solutions are smooth, and the dual problem is strongly concave. In this talk, we will discuss quadratically regularized optimal transport (QOT), which uses the $L^2$ entropy. However, the dual of QOT is neither strongly concave, nor are its dual solutions necessarily smoother than those of OT. This raises several natural questions: How quickly does the QOT cost converge to the classical OT cost? Are there efficient algorithms? Does QOT suffer from the curse of dimensionality? We review recent results answering these questions.
17:00-17:30
Katharina Schuh (Technische Universität Wien)
In this talk we study the kinetic Langevin dynamics and several associated discretization schemes, including the Euler discretization and splitting methods. We focus on their contraction behaviour and their efficiency in sampling a given probability distribution in $\mathbb{R}^d$ with non-convex potential. We show quantitative contraction results in $L^1$-Wasserstein distance for both the continuous dynamics and the discretization schemes. These results are based on a carefully tailored distance function and an appropriate coupling construction. Finally, we quantify the error in the $L^1$-Wasserstein distance between the invariant measure of the continuous dynamics and of the discretization schemes.
Friday, 10 July
14:00-14:30
Quentin Merigot (Université Paris-Saclay)
The stability of optimal transport maps has been the focus of significant recent research, driven in part by applications in numerical analysis, statistics, sampling theory and data science (the “linearized optimal transport” framework). From a PDE perspective, this stability question amounts to quantifying the well-posedness of the (real) Monge-Ampère equation associated to optimal transport, and to establish continuous dependence on the data even when solutions lacks regularity. In this presentation I will present a recent stability inequality with a sharp exponent.
14:30-15:00
Eva Kopfer (Universität Bonn)
In this talk we examine the large-scale behavior of dynamical optimal transport on graphsℝn. Our central theorem is a homogenization result that characterizes the effective behavior of the discrete problems in terms of a continuous optimal transport problem.
15:00-15:30
Dejan Slepcev (Carnegie Mellon University)
Sampling from a probability measure known only through its unnormalized density is a fundamental problem, which becomes especially challenging in high dimensions. We will discuss interacting-particle approaches to sampling based on gradient flows of the Kullback–Leibler (KL) divergence. Such flows, including the Fokker–Planck equation and Stein Variational Gradient Descent (SVGD), evolve a distribution towards a desired target measure. We will focus on a new geometry on the space of measures, named Radon-Wasserstein geometry. The resulting gradient flows of the KL divergence have the following remarkable properties: they can be accurately approximated in high dimensions based on interacting particles and the computational complexity of each step is linear in the number of particles and the dimension. We will discuss theoretical results such as well-posedness of the gradient flow, its convergence towards the target measure, and convergence of algorithms. We also describe an affine-invariant variant of the metric, designed for target distributions with large condition numbers while retaining the favorable properties of the original framework.
15:30-16:00
Olga Mula (University of Vienna)
16:00-16:30 Coffee Break
16:30-17:00
Lauren Conger (Caltech)
We study the evolution variational inequality (EVI) driven by a bifunction on a metric space, providing conditions for existence of solutions, establishing solution properties, and characterizing asymptotic behavior. A motivating application is the setting of multispecies and minimax coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a penalized bifunction, generalizing the classical minimizing movement scheme for single species gradient descent. In the coupled gradient flow case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete VMS solutions via two approaches: a primal approach using a metric-space version of the KKM theorem and compactness of superlevel sets, and a dual approach using two structural conditions—barycentric convexity and interaction dissipativity—which together provide a new zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure. This is joint work with Franca Hoffmann and Giuseppe Savare.
17:00-17:30
Andrea Natale (INRIA Saclay)
Wasserstein barycenters with signed coefficients provide a variational notion of extrapolation in Wasserstein space. Such signed barycenters arise in applications ranging from statistical data analysis to the discretization of PDEs, but the presence of negative weights makes the problem non-convex and challenging both to analyze and to solve numerically. Using Toland duality, we derive a precise characterization of signed barycenters through convex-order conditions and uncover a hidden convex structure in the problem. Building on this, we develop a particle-based algorithm for their computation, combining entropic regularization with DC programming.
17:30-18:00
Thomas Gallouët (INRIA Saclay)
Laguerre tessellations offer an efficient way to parameterize a large class of convex partitions of Euclidean space using only a set of points and scalar weights. For this reason, they have become popular in computational geometry, imaging and numerical analysis, both as a modeling and a discretization tool. In this talk we will discuss the problem of reconstructing a Laguerre tessellation with prescribed cell volumes from the barycenters of its cells.
We will discuss a geometric interpretation of this problem in terms of the set of discrete measures dominated in convex order by an absolutely continuous measure. In particular,
the reconstruction problem can be solved approximately by computing a Wasserstein projection onto this set. More generally this method can also be applied to fit a Laguerre tessellation to an arbitrary set of barycenters.
Saturday, 11 July
14:00-14:30
Brendan Pass (University of Alberta)
We present a primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions. Even in the two-marginal setting, this formulation applies to cost functions not covered by the classical dynamical approach of Benamou-Brenier. Our dynamical formulation yields a convex optimization problem, enabling the use of convex optimization tools to find quasi-Monge solutions of the static multi-marginal problem for translation-invariant costs. We illustrate our results numerically with proximal splitting methods. This is joint work with Yair Shenfeld.
14:30-15:00
Julio Backhoff-Veraguas (University of Vienna)
Martingale Schrödinger Bridges, introduced by Nutz and Wiesel, are the minimizers of a relative entropy cost function over the class of martingales in one step with prescribed initial and terminal marginals. One can therefore say that they are the most independent martingale coupling given these marginals.
In this talk we clarify the structure of martingale Schrödinger bridges in the irreducible case. In particular, we show that they can be obtained by means of a classical entropic optimal transport problem, where the initial distribution is replaced by a possibly different auxiliary measure, and show how to find the latter. We further identify the continuous-time analogue of this problem, and find a novel connection with the field of stochastic filtering.
15:00-15:30
Gudmund Pammer (Graz University of Technology)
The Schrödinger-Bass problem is a semimartingale optimal transport problem connecting the celebrated Schrödinger bridge with the Bass martingale problem. In this talk, I will explain how the dynamic Schrödinger-Bass problem admits an equivalent static formulation as a weak optimal transport problem with an explicit cost. This reformulation yields existence and duality results, provides a structural characterization of the optimal semimartingales, and naturally leads to a Sinkhorn-type iteration for computation.
15:30-16:00
Luca Tamanini (Università Cattolica Del Sacro Cuore)
In this talk we study stability of optimizers and convergence of Sinkhorn’s algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals. When employed in the analysis of Sinkhorn’s algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. (based on a joint work with A. Chiarini, G. Conforti, and G. Greco)
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Mathias Beiglböck (University of Vienna)
We establish a Brenier theorem for iterated Wasserstein spaces. Specifically, for a separable Hilbert space $H$ and $N\ge 1$, we construct a full-support probability $\Lambda\in P_2^N(H)= P_2(… P(H)…)$ 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 type. In the first non-classical case $N=2$ we show that optimal transports are given as the push-forward by the $W_2$-gradient (or Lions’ derivative) of an $L$-convex function. To establish the result for general $N$ we develop new adapted notions of Lions’ lift, $L$-convexity and Lions’ derivative. A key idea is a new identification between optimal-transport $c$-conjugation (with $c$ given by maximal covariance) and classical convex conjugation on the lift.
A primary motivation comes from the adapted Wasserstein distance $AW_2$: our results yield a first Brenier theorem for $AW_2$ and characterize $AW_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.
17:30-18:00
Giovanni Conforti (University of Padova)
We consider the problem of finding the best Gaussian approximation of a given diffusion process, possibly of Mc-Kean Vlasov type. This optimization task can be interpreted as performing variational inference for path measures. We derive a set of Kalman-McKean Vlasov equations for the optimal Gaussian process, provide a bound on the approximation error, and study its long-time behavior. Our analysis builds on an interpretation of optimizers as projected Newton’s laws on Wasserstein space, which is of independent interest. Joint work with L.P. Chaintron
Posters
![[PZ13 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Pierre-Cyril Aubin
(Ecole des Ponts ParisTech)
Since a long time, the ubiquitous costs in optimal transport were the Wasserstein distances $W_p$. Being distances, they had no need to be debiased. However with entropic regularizations, the self-cost is no longer null. This justifies debiasing, e.g. by defining the Sinkhorn divergence. Similarly, in quantum optimal transport, it is possible to define analogues of the Wasserstein distances from classical OT, however the self-cost is also no longer null. We detail a widely applicabl inf-representation of costs which allows to guarantee that their debiased versions are positive.
Mia Busuladzic-Begic
(Brown University)
Forecasting in the Wasserstein space $(\mathcal{P}2(\mathbb{R}^d), W_2)$ requires embedding infinite-dimensional, non-Euclidean dynamics into a tractable computational framework. We address this by combining the linearized optimal transport (LOT) embedding with reservoir computing. Given a trajectory $(\mu_t){t \ge 0} \subset \mathcal{P}2(\mathbb{R}^d)$, we embed each measure into the Hilbert space $L^2(\sigma; \mathbb{R}^d)$ via the optimal transport map from a fixed reference $\sigma$, and train an echo state network (ESN) on LOT velocities $\Delta_t \;=\; T\sigma^{\mu_{t+1}} – T_\sigma^{\mu_t}$.
]
Our first contribution is a dynamical justification for this choice: when $(\mu_t)$ satisfies the continuity equation $\partial_t \mu_t +
abla \cdot (v_t \mu_t) = 0$ and the reference is (approximately) Lagrangian-compatible, the LOT lift $u_t = T_\sigma^{\mu_t}$ evolves by $\dot{u}_t = v_t \circ u_t,$ so LOT velocities coincide with pullbacks of the Eulerian velocity field and preserve kinetic energy in $L^2(\sigma)$. We further provide a finite-sample error budget that separates LOT linearization error (controlled by a compatibility parameter), statistical transport-map estimation error, and reference discretization error.
Our second contribution establishes learnability: under the echo state property, ESNs uniformly approximate the induced Koopman action on LOT observables over compact trajectory sets, and reconstruction by pushforward is stable, yielding Wasserstein error control from $L^2(\sigma)$ prediction error.
Extensive experiments on benchmark measure-valued dynamical systems demonstrate that LOT-velocity reservoir forecasting outperforms direct particle-level forecasting under certain system conditions. We test across reference choices, particle counts, and reservoir scales, with order-of-magnitude improvements in Wasserstein error on several benchmarks.
William Ford
(École Polytechnique)
We study the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers).
On the qualitative side, we prove that when one marginal measure’s support is rectifiably connected (path-connected by rectifiable paths), the optimal dual potentials are unique up to a constant. This result is applicable even when both marginal measures are concentrated on lower-dimensional subsets of the ambient space, and in cases where optimal potentials are nowhere differentiable on the supports of the marginals.
On the quantitative side, we control the diameter of the set of optimal dual potentials by the Hausdorff distance between the support of one of the marginal measures and a regular connected set. In this way, we quantify the extent to which optimisers are almost unique when the support of one marginal measure is almost connected. This is a consequence of a novel characterisation of the set of optimal dual potentials as the intersection of an explicit family of half-spaces. These results apply directly to common situations in applications: they say that when one measures’ support is a fine discretisation of a connected set, or is an empirical measure with a large number of sampled points, the dual solutions are very close to being unique, the diameter of the set of optimisers is quantitatively small.
Ting Gao
(Huazhong University Of Science And Technology)
We introduce a framework for analyzing topological tipping in time evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, homological and higher‐order relations into one metric, its global scalar distance can obscure transient, localized structural reorganizations during dynamic phase transitions. To overcome this limitation, we present a hierarchical dynamic evaluation framework driven by a novel topological and hypergraph reconstruction strategy. Instead of directly interpolating abstract network parameters, our method interpolates the underlying spatial geometry and rigorously re-computes the valid topological structures, ensuring physical fidelity. Along this geodesic, we introduce a set of multi-scale indicators: macroscopic metrics (Topological Distortion and Persistence Entropy) to capture global shifts, and a novel mesoscopic dual-perspective Hypergraph Entropy (nodeperspective and edge-perspective) to detect highly sensitive, asynchronous local rewirings. We further propagate the cycle-level entropy change onto individual vertices to form a point-level topological field. Extensive evaluations on physical dynamical systems (Rayleigh-Van der Pol limit cycles, Double-Well cluster fusion), high-dimensional biological aggregation (D’Orsogna model), and longitudinal stroke fMRI data demonstrate the utility of combining transport‐based alignment with multi‐scale entropy diagnostics for dynamic topological analysis.
John Hopper
(University Of California, Los Angeles)
The sliced Wasserstein metric has major computational advantages over the standard Wasserstein metric, and as such has seen much use in applications using Wasserstein metrics. Yet the geometric structure of these metrics and how they differ from the original is less studied. It is known almost since the inception of the sliced metric that translations and dilations give rise to constant speed (extrinsic) geodesics for both metrics. Later, in the work of Kitagawa and Takatsu that the sliced Wasserstein metric is not a length space, that is not every inartistic geodesic is extrinsic. This work explores the first examples of extrinsic geodesics in the sliced Wasserstein metrics other than dilations and translations. These geodesics are not geodesic with respect to the Wasserstein metric and demonstrate the different geometric properties of these metric spaces. One notable application of these curves is that they can be used to prove the non-equivalence of the sliced Wasserstein and Wasserstein metrics in the cases previously unknown in the literature, giving the proof that these metrics are not bi-Lipchitz equivalent in any dimension greater than 1.
Maria Oprea
(Cornell University)
Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this project, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric — both in the design of the loss function and in the optimization process — significantly impacts the stability and properties of the optimizer.
Maximilian Penka
(UMLP / CNRS)
We consider nonlinear approximation methods for parametrized partial differential equations based on optimal transport, building on recent work on nonlinear reduced basis methods using mixture Wasserstein barycenters [1]. The approach is motivated by problems arising in electronic structure calculations, where solutions can naturally be described as probability measures or mixtures of simple atomic densities.
In contrast to classical projection-based reduction techniques, the method operates directly on a nonlinear manifold of probability measures, enabling interpolation and approximation in a metric induced by optimal transport rather than in linear function spaces. This perspective is particularly well-suited for problems where solutions exhibit localization, symmetry, or low-dimensional structure in parameter space.
We focus on mixture-based Wasserstein-type distances, which allowto construct barycentric approximations while retaining a computationally tractable structure. Reduced models are obtained by combining a small number of precomputed solutions through transport-based interpolation.
We are currently investigating the mathematical and numerical foundations of this approach, including stability, approximation properties, and the construction of computationally efficient and mathematically well-founded reduced models.
[1] Dalery, M., Dusson, G., Ehrlacher, V. et al. Nonlinear Reduced Basis using Mixture Wasserstein Barycenters: Application to an Eigenvalue Problem Inspired from Quantum Chemistry. Constr Approx (2026).
Moritz Piening
(TU Berlin)
Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes. Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space of square-integrable functions. For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between 1d meta-measures via infinite-dimensional projections, parametrized by Gaussian processes. Combining this 1d construction with classical integration over the Euclidean unit sphere yields the double-sliced Wasserstein (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.
Viktor Stein
(TU Berlin)
Most f-divergences like, e.g., the Kullback-Leibler divergence, are infinite between two measures with disjoint support. Aiming to obtain a better functional for descent schemes used for example in generative modelling, we propose to remedy this property while still keeping as much of the “f-divergence geometry” as possible. We achieve this by regularizing the f-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel K. We use the kernel mean embedding to show that this regularization can be rewritten as the Moreau envelope of some function on the reproducing kernel Hilbert space associated to K. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to analyze the MMD-regularized f-divergences, particularly their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized f-divergences. We prove that flows starting at an empirical measure stay empirical and that the explicit time discretization of these flows can be tractably computed. Our general theory allows the unified treatment of f-divergences with infinite and finite recession constants and also so-called “tight” f-divergences. Finally, we will present numerical examples to visualize the impact of the choice of kernel and regularization strength.
This is joint work with Sebastian Neumayer, Nicolaj Rux and Gabriele Steidl, published in Analysis and Applications.
Guillaume Wang
(New York University)
The Sinkhorn algorithm is the de facto standard method for numerically solving entropy-regularized optimal transport problems over finite sets. In this work, we identify and explain a phenomenon arising when Sinkhorn is applied with a small regularization parameter $\tau$: the evolution of the variables $(f_k, g_k)$ is approximately piecewise-linear, and their increments belong to a finite set determined solely by the marginals, independent of the cost matrix. We prove that as $\tau \to 0$, the rescaled Sinkhorn iterates $(f_{t/\tau}, g_{t/\tau})$ indeed converge to a continuous-time curve consistent with the above observations, and we describe the limiting dynamics explicitly. Furthermore, we demonstrate through numerical experiments that some features of the limiting dynamics can be observed as soon as $\tau \log (m \vee n) = o(1)$, where $m$ and $n$ are the domains’ cardinalities—precisely the regime advocated by the existing literature on computational optimal transport (Altschuler et al., 2019).