Research interests
My main interest lies in optimal transport and the geometry of the Wasserstein space, convex analysis and optimization, calculus of variations and Bayesian statistics. Below is a list of publications organized by research themes:
- Optimal curves valued in the space of measures.
- Optimal mappings valued in the space of measures.
- Numerical solution to the dynamical optimal transport problem.
- Optimal transport distances in Bayesian Nonparametrics.
- Optimal transport and calculus of variations in data science.
You can also find on this webpage slides of the talks I have given.
Optimal curves valued in the space of measures
Optimal transport provides a natural way to interpolate between probability distributions, generating geodesics (that is, shortest paths) in the so-called Wasserstein space. I have studied some variations of this problem where one not only looks for the shortest path, but adds additional terms in the objective functional. Penalization of congestion yields model of fluid dynamics and is connected to Mean Field Games; allowing for growth terms leads to unbalanced optimal transport.
- Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion. Aymeric Baradat and HL. 2021. Accepted paper: Astérisque.
- New estimates on the regularity of the pressure in density-constrained Mean Field Games. HL and Filippo Santambrogio. 2019. Accepted paper: Journal of the London Mathematical Society.
- Optimal density evolution with congestion: L infinity bounds via flow interchange techniques and applications to variational Mean Field Games. HL and Filippo Santambrogio. 2018. Accepted paper: Communications in PDEs.
- Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation. HL. 2017. Accepted paper: Calculus of Variations and PDEs.
Optimal mappings valued in the space of measures
An extension of the geodesic problem is to consider not curves valued in the Wasserstein space, but mappings: that is, probability measures depending on more than one parameter. I have worked on a definition of Dirichlet energy for such mappings proposed by Brenier, studied its minimizers (the harmonic mappings) and with collaborators we proposed an application to the theory of elasticy. This is related to analysis in non smooth spaces and the problem of harmonic metric valued mappings.
- Lifting functionals defined on maps to measure-valued maps via optimal transport. HL. 2023. Preprint.
- Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps. HL, Léonard Monsaingeon, Luca Tamanini and Dmitry Vorotnikov. 2021. Accepted paper: Calculus of Variations and PDEs.
- Hidden convexity in a problem of nonlinear elasticity. Nassif Ghoussoub, Young-Heon Kim, HL and Aaron Zeff Palmer. 2021. Accepted paper: SIAM Journal on Mathematical Analysis.
- Harmonic mappings valued in the Wasserstein space. HL. 2019. Accepted paper: Journal of Functional Analysis.
Numerical solution to the dynamical optimal transport problem
I have worked on the dynamical optimal transport problem (a.k.a. Benamou-Brenier formulation) and its numerical solution. I have proposed a new discretization of the problem on surfaces, and also done the analysis of the convergence of solutions to the discrete problems to the continuous one when the spatial and temporal step size vanish.
- Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation. Sadashige Ishida and HL. 2023. Submitted paper.
- Unconditional convergence for discretizations of dynamical optimal transport. HL. 2021. Accepted paper: Mathematics of Computations.
- Dynamical Optimal Transport on Discrete Surfaces (supplemental, code). HL, Sebastian Claici, Edward Chien and Justin Solomon, 2018. Accepted paper: SIGGRAPH Asia 2018.
Optimal transport distances in Bayesian Nonparametrics
With collaborators in Bayesian statistics, we have been using optimal transport to define distances between Completely Random Measures. With the help of such distances, we are able to define a tractable index of dependence between Completely Random Vectors and evaluate the impact of the prior in a nonparametric setting.
- Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity. Marta Catalano and HL. 2024. Preprint.
- Merging Rate of Opinions via Optimal Transport on Random Measures. Marta Catalano and HL. 2023. Preprint.
- A Wasserstein index of dependence for random measures. Marta Catalano, HL, Antonio Lijoi and Igor Prünster. 2021. Accepted paper: Journal of the American Statistical Association.
Optimal transport and calculus of variations in data science
Below is a list of projects (unrelated one to the other) involving optimal transport and calculus of variations in data science, understood in a broad sense. In particular I have worked on the trajectory inference problem (reconstructing the law of a stochastic process from samples of its temporal marginals) and connected it to entropic optimal transport and the Schrödinger problem.
- The flow map of the Fokker-Planck equation does not provide optimal transport. HL and Filippo Santambrogio. 2022. Accepted paper: Applied Mathematics Letters.
- Towards a mathematical theory of trajectory inference. HL*, Stephen Zhang*, Young-Heon Kim and Geoffrey Schiebinger. 2021. Accepted paper: Annals of Applied Probability.
- Total Variation Isoperimetric Profiles. Daryl DeFord, HL, Zachary Schutzman and Justin Solomon. 2019. Accepted paper: SIAM Journal on Applied Algebra and Geometry.
Slides
Below are the slides of some talks I gave. As they were designed to go with an oral presentation, they may not be self sufficient.
- What distance to use between probabilities over probabilities?
- Lifting functionals defined on maps to measure-valued maps via optimal transport.
- Quantifying the merging of opinions in Bayesian nonparametrics via optimal transport.
- Wasserstein index of dependence in Bayesian nonparametrics: version for statisticians, version for analysts.
- A probabilistic view on unbalanced on optimal transport.
- Using optimal transport for trajectory inference. About the work on trajectory inference. It presents also some results of the project on Waddington OT that I was not a part of.
- Hidden convexity in a problem of non linear elasticity (also in French).
- Dynamical Optimal Transport: discretization and convergence. Both about numerics and about proof of convergence of the discretizations.
- Dynamical Optimal Transport on Discrete Surfaces. The slides of the SIGGRAPH Asia talk.
- Harmonic mappings valued in the Wasserstein. I've sometimes presented different results in the last part. These slides contain numerical computations of harmonic mappings.
- Variational Mean Field Games: on estimates on the density and the pressure. This is about the two articles on Mean Field Games I wrote with Filippo Santambrogio.
PhD
Between 2016 and 2019, I worked under the supervision of Filippo Santambrogio to complete a PhD. My PhD manuscript can be found here and the slides of my defense here.
Undergraduate works
You can find below works that I did before starting my PhD.
- My master thesis in history of science (September 2016, in French). It was realized under the supervision of Laurent Mazliak and was about the history of the teaching of probability calculus and statistics in France.
- My master thesis in mathematics (September 2015, in French). It was realized under the supervision of Filippo Santambrogio.
- My internship report of Master 1 (November 2014). It summarizes the work done during an intership at CalTech (United States of America) under the supervision of Oscar Bruno and Edwin Jimenez about stability problems in numerical analysis.
- The research conducted during my internship at CEA Saclay (June and July 2013) under the supervision of Grégoire De Loubens led to a publication: Mechanical magnetometry of Cobalt nanospheres deposited by focused electron beam at the tip of ultra-soft cantilevers. HL, Vladimir Naletov, Olivier Klein, Grégoire De Loubens, Laura Casado, and José Maria De Teresa. 2014. Nanofabrication, 1(1).