Published and Accepted Papers:
Optimal transportation in a discrete setting [arXiv:1911.00574]
I studied the regularity of Kantorovich potentials for the optimal transportation problem between measures that are not necessarily absolutely continuous, but that behave like absolutely continuous measures up to a certain scale. Examples of these are discretizations of absolutely continuous measures on grids.
Local regularity result for an optimal transportation problem with rough measures on the plane
(PE Jabin, A Mellet, M Molina-Fructuoso, Journal of Functional Analysis, Volume 281, Issue 2, 2021)
Tukey Depths and Hamilton-Jacobi Differential Equations [arXiv:2104.01648]
In colaboration with Professor Ryan Murray, I developed a PDE-based formulation for the Tukey depth of a reasonably smooth probability density. We showed that the viscosity solution of this equation is well-posed in any dimension and that, under certain assumptions, it coincides with the Tukey depth of the distribution in 2 dimensions.
Tukey Depths and Hamilton-Jacobi Differential Equations
(Martin Molina-Fructuoso, Ryan Murray, SIAM Journal on Mathematics of Data Science, Volume 4, Issue 2, 2022)
Eikonal depth: an optimal control approach to statistical depths [arXiv:2201.05274]
Acceleration and implicit regularization in phase retrieval [arXiv:2311.12888 ]
A description of my research on optimization and PDE-based variational methods for machine learning, and on optimal transportation, as well as future research directions can be found in my Research Statement .