Research

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. The years indicated refer to the first submission of the work.

Optimal curves and mappings 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.

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.

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.

Optimal transport in Bayesian statistics

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. I have also used optimal transport and variational approaches to study the sampling problem with coordinate-wise methods, such as Coordinate Ascent Variational Inference and Gibbs sampling.

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.

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.