@article{20050, abstract = {We prove upper bounds on the L∞-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the L∞-Wasserstein metric and the relative L∞-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher’s infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions.}, author = {Khudiakova, Kseniia and Maas, Jan and Pedrotti, Francesco}, issn = {1050-5164}, journal = {The Annals of Applied Probability}, number = {3}, pages = {1913--1940}, publisher = {Institute of Mathematical Statistics}, title = {{L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model}}, doi = {10.1214/25-aap2162}, volume = {35}, year = {2025}, } @unpublished{20569, abstract = {This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy d between two probability distributions in position and velocity states, which is reminiscent of the 2-Wasserstein distance, but of second-order nature. We construct d in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon T. Second, we further optimise over the time horizon T > 0. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of d. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of d holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by d, and identify solutions to Vlasov's equations with curves of measures satisfying a certain d-absolute continuity condition. One consequence is an explicit formula for the d-derivative of such curves.}, author = {Brigati, Giovanni and Maas, Jan and Quattrocchi, Filippo}, booktitle = {arXiv}, keywords = {optimal transport, kinetic theory, second-order discrepancy, Vlasov equation, Wasserstein splines.}, title = {{Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures}}, doi = {10.48550/arXiv.2502.15665}, year = {2025}, } @article{17143, abstract = {This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn. .}, author = {Dello Schiavo, Lorenzo and Maas, Jan and Pedrotti, Francesco}, issn = {1088-6850}, journal = {Transactions of the American Mathematical Society}, number = {6}, pages = {3779--3804}, publisher = {American Mathematical Society}, title = {{Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces}}, doi = {10.1090/tran/9156}, volume = {377}, year = {2024}, } @inproceedings{18897, abstract = {Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time T1 by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires T1→∞. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as T1 diverges; from a practical viewpoint, a large T1 increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time T1. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the L2 loss on the score approximation, which is the quantity minimized in practice.}, author = {Pedrotti, Francesco and Maas, Jan and Mondelli, Marco}, booktitle = {Transactions on Machine Learning Research}, issn = {2835-8856}, title = {{Improved convergence of score-based diffusion models via prediction-correction}}, year = {2024}, } @book{18899, abstract = {The flourishing theory of classical optimal transport concerns mass transportation at minimal cost. This book introduces the reader to optimal transport on quantum structures, i.e., optimal transportation between quantum states and related non-commutative concepts of mass transportation. It contains lecture notes on classical optimal transport and Wasserstein gradient flows dynamics and quantum optimal transport quantum couplings and many-body problems quantum channels and qubits These notes are based on lectures given by the authors at the "Optimal Transport on Quantum Structures" School held at the Erdös Center in Budapest in the fall of 2022. The lecture notes are complemented by two survey chapters presenting the state of the art in different research areas of non-commutative optimal transport.}, editor = {Maas, Jan and Rademacher, Simone Anna Elvira and Titkos, Tamás and Virosztek, Daniel}, isbn = {9783031504655}, issn = {2947-9460}, publisher = {Springer Nature}, title = {{Optimal Transport on Quantum Structures}}, doi = {10.1007/978-3-031-50466-2}, volume = {29}, year = {2024}, } @article{17282, abstract = {Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric gαβ on M such that Yβ=gαβXα ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.}, author = {Brooks, Morris and Maas, Jan}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {6}, publisher = {Springer Nature}, title = {{Characterisation of gradient flows for a given functional}}, doi = {10.1007/s00526-024-02755-z}, volume = {63}, year = {2024}, } @unpublished{17350, abstract = {Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time $T_1$ by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires $T_1\to\infty$. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as $T_1$ diverges; from a practical viewpoint, a large $T_1$ increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time $T_1$. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the $L^2$ loss on the score approximation, which is the quantity minimized in practice.}, author = {Pedrotti, Francesco and Maas, Jan and Mondelli, Marco}, booktitle = {arXiv}, title = {{Improved convergence of score-based diffusion models via prediction-correction}}, doi = {10.48550/arXiv.2305.14164}, year = {2024}, } @unpublished{17352, abstract = {We prove upper bounds on the $L^\infty$-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the $L^\infty$-Wasserstein metric and the relative $L^\infty$-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher's infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions.}, author = {Khudiakova, Kseniia and Maas, Jan and Pedrotti, Francesco}, booktitle = {arXiv}, title = {{L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model}}, doi = {10.48550/arXiv.2402.04151}, year = {2024}, } @article{12959, abstract = {This paper deals with the large-scale behaviour of dynamical optimal transport on Zd -periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ -convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {5}, publisher = {Springer Nature}, title = {{Homogenisation of dynamical optimal transport on periodic graphs}}, doi = {10.1007/s00526-023-02472-z}, volume = {62}, year = {2023}, } @article{11700, abstract = {This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.}, author = {Erbar, Matthias and Forkert, Dominik L and Maas, Jan and Mugnolo, Delio}, issn = {1556-181X}, journal = {Networks and Heterogeneous Media}, number = {5}, pages = {687--717}, publisher = {American Institute of Mathematical Sciences}, title = {{Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph}}, doi = {10.3934/nhm.2022023}, volume = {17}, year = {2022}, } @article{11739, abstract = {We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.}, author = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence}, number = {4}, pages = {4297--4333}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}}, doi = {10.1137/21M1410968}, volume = {54}, year = {2022}, } @article{10023, abstract = {We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.}, author = {Karatzas, Ioannis and Maas, Jan and Schachermayer, Walter}, issn = {1526-7555}, journal = {Communications in Information and Systems}, keywords = {Markov Chain, relative entropy, time reversal, steepest descent, gradient flow}, number = {4}, pages = {481--536}, publisher = {International Press}, title = {{Trajectorial dissipation and gradient flow for the relative entropy in Markov chains}}, doi = {10.4310/CIS.2021.v21.n4.a1}, volume = {21}, year = {2021}, } @article{6358, abstract = {We study dynamical optimal transport metrics between density matricesassociated to symmetric Dirichlet forms on finite-dimensional C∗-algebras. Our settingcovers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, andspectral gap estimates.}, author = {Carlen, Eric A. and Maas, Jan}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, number = {2}, pages = {319--378}, publisher = {Springer Nature}, title = {{Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems}}, doi = {10.1007/s10955-019-02434-w}, volume = {178}, year = {2020}, } @unpublished{10022, abstract = {We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in R^d and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.}, author = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo}, booktitle = {arXiv}, title = {{Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}}, doi = {10.48550/arXiv.2008.10962}, year = {2020}, } @article{7573, abstract = {This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such metrics appear naturally in discretisations of -gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to , unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo}, issn = {0021-7824}, journal = {Journal de Mathematiques Pures et Appliquees}, number = {7}, pages = {204--234}, publisher = {Elsevier}, title = {{Homogenisation of one-dimensional discrete optimal transport}}, doi = {10.1016/j.matpur.2020.02.008}, volume = {139}, year = {2020}, } @article{8758, abstract = {We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.}, author = {Maas, Jan and Mielke, Alexander}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, number = {6}, pages = {2257--2303}, publisher = {Springer Nature}, title = {{Modeling of chemical reaction systems with detailed balance using gradient structures}}, doi = {10.1007/s10955-020-02663-4}, volume = {181}, year = {2020}, } @article{71, abstract = {We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric 𝕎2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of 𝕎2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, number = {3}, pages = {2759--2802}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Scaling limits of discrete optimal transport}}, doi = {10.1137/19M1243440}, volume = {52}, year = {2020}, } @article{73, abstract = {We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.}, author = {Erbar, Matthias and Maas, Jan and Wirth, Melchior}, issn = {0944-2669}, journal = {Calculus of Variations and Partial Differential Equations}, number = {1}, publisher = {Springer}, title = {{On the geometry of geodesics in discrete optimal transport}}, doi = {10.1007/s00526-018-1456-1}, volume = {58}, year = {2019}, } @article{956, abstract = {We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C⁎-algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.}, author = {Carlen, Eric and Maas, Jan}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, number = {5}, pages = {1810 -- 1869}, publisher = {Academic Press}, title = {{Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance}}, doi = {10.1016/j.jfa.2017.05.003}, volume = {273}, year = {2017}, } @inbook{649, abstract = {We give a short overview on a recently developed notion of Ricci curvature for discrete spaces. This notion relies on geodesic convexity properties of the relative entropy along geodesics in the space of probability densities, for a metric which is similar to (but different from) the 2-Wasserstein metric. The theory can be considered as a discrete counterpart to the theory of Ricci curvature for geodesic measure spaces developed by Lott–Sturm–Villani.}, author = {Maas, Jan}, booktitle = {Modern Approaches to Discrete Curvature}, editor = {Najman, Laurent and Romon, Pascal}, isbn = {9783319580012}, pages = {159 -- 174}, publisher = {Springer}, title = {{Entropic Ricci curvature for discrete spaces}}, doi = {10.1007/978-3-319-58002-9_5}, volume = {2184}, year = {2017}, }