@article{molina-fructuoso_murray_2022, title={Tukey Depths and Hamilton-Jacobi Differential Equations}, volume={4}, ISSN={["2577-0187"]}, url={https://doi.org/10.1137/21M1411998}, DOI={10.1137/21M1411998}, abstractNote={Widespread application of modern machine learning has increased the need for robust statistical algorithms. This work studies one such fundamental statistical concept known as the Tukey depth. We study the problem in the continuum (population) limit. In particular, we formally derive the associated necessary conditions, which take the form of a first-order partial differential equation which is necessarily satisfied at points where the Tukey depth is smooth. We discuss the interpretation of this formal necessary condition in terms of the viscosity solution of a Hamilton--Jacobi equation, but with a nonclassical Hamiltonian with discontinuous dependence on the gradient at zero. We prove that this equation possesses a unique viscosity solution and that this solution always bounds the Tukey depth from below. In certain cases we prove that the Tukey depth is equal to the viscosity solution, and we give some illustrations of standard numerical methods from the optimal control community which deal directly with the partial differential equation. We conclude by outlining several promising research directions both in terms of new numerical algorithms and theoretical challenges.}, number={2}, journal={SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE}, author={Molina-Fructuoso, Martin and Murray, Ryan}, year={2022}, pages={604–633} }
@article{jabin_mellet_molina-fructuoso_2021, title={Local regularity result for an optimal transportation problem with rough measures in the plane}, volume={281}, ISSN={["1096-0783"]}, DOI={10.1016/j.jfa.2021.109041}, abstractNote={We investigate the properties of convex functions in R 2 that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampère equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a C 1 regularity result up to the discrete scale for the Legendre transform. Our result applies in particular to any Kantorovich potential associated to an optimal transportation problem between two measures that are (possibly only locally) sums of uniformly distributed Dirac masses. The proof relies on novel explicit estimates directly based on the optimal transportation problem, instead of the Monge-Ampère equation.}, number={2}, journal={JOURNAL OF FUNCTIONAL ANALYSIS}, author={Jabin, P-E and Mellet, A. and Molina-Fructuoso, M.}, year={2021}, month={Jul} }