Acceleration and implicit regularization in phase retrieval
Tyler Maunu, Martin Molina-Fructuoso, IASTATS 2024 (International Conference on Artificial Intelligence and Statistics) [arXiv:2311.12888]
We study accelerated optimization methods for the Gaussian phase retrieval problem, which is nonconvex. We prove that Polyak's and Nesterov's gradient accelerated methods exhibit similar implicit regularization properties to gradient descent and present numerical experiments, which agree with these results.


Eikonal depth: an optimal control approach to statistical depths
Martin Molina-Fructuoso, Ryan Murray, Foundations of Data Science (Springfield, Mo.), Early Access, 2024 [arXiv:2201.05274]
We propose a notion of deph for probability distributions based on optimal trajectories and eikonal equations. We show that this concept is extendable to more general settings such as point clouds. In particular, we illustrate how our depth defines an ordering in subsets of MNIST corresponding to a given number.

Tukey Depths and Hamilton-Jacobi Differential Equations
Martin Molina-Fructuoso, Ryan Murray, SIAM Journal on Mathematics of Data Science, Volume 4, Issue 2, 2022 [arXiv:2104.01648]
In colaboration with Professor Ryan Murray, I develop a PDE-based formulation for the Tukey depth of a reasonably smooth probability density. We show 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.

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 [arXiv:1911.00574]
I study 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.

A characteristics-based approach to computing Tukey depths
Manuscript in preparation


Research Statement:
A description of my research on optimization and PDE-based variational methods for machine learning and data science, and on optimal transportation, as well as some ideas for future research can be found in my Research Statement .