arXiv subjects

Rene Cabrera

Publications and source records attributed to Rene Cabrera.

4 recordsLinked to original sources

arXiv2026

A minimax Bilinear Transport Problem and Nash-Monge-Kantorovich Maps

We study a min-max bilinear transport problem arising from a two-player zero-sum game with quadratic kinetic and interaction costs. Starting from a dynamic path space formulation, we establish existence of minimax and maximin plans and prove a minimax theorem. We show that the equilibrium induces a finite-dimensional stationary problem via an endpoint cost on transport plans, which is well defined below a critical interaction strength and yields a Nash equilibrium over couplings. In the quadratic interaction case, we derive an explicit endpoint cost and a dual formulation. The resulting Nash-Monge-Kantorovich (NMK) plans admit Monge solutions, recovering classical structures in optimal transport, with optimal maps given by gradients of convex or concave functions when they exist. Our analysis highlights duality and cyclical (anti-)monotonicity for nonstandard costs and links the equilibrium maps to coupled nonlinear PDEs, bridging optimal transport, zero-sum games, and Monge-Ampere-type equations.

math.AP↗

arXiv2024

GAN: Dynamics

We study quantitatively the overparametrization limit of the original Wasserstein-GAN algorithm. Effectively, we show that the algorithm is a stochastic discretization of a system of continuity equations for the parameter distributions of the generator and discriminator. We show that parameter clipping to satisfy the Lipschitz condition in the algorithm induces a discontinuous vector field in the mean field dynamics, which gives rise to blow-up in finite time of the mean field dynamics. We look into a specific toy example that shows that all solutions to the mean field equations converge in the long time limit to time periodic solutions, this helps explain the failure to converge.

math.AP↗

arXiv2023

Regularization estimates of the Landau-Coulomb diffusion

The Landau-Coulomb equation is an important model in plasma physics featuring both nonlinear diffusion and reaction terms. In this manuscript we focus on the diffusion operator within the equation by dropping the potentially nefarious reaction term altogether. We show that the diffusion operator in the Landau-Coulomb equation provides a much stronger L^1 to L^\infty rate of regularization than its linear counterpart, the Laplace operator. The result is made possible by a nonlinear functional inequality of Gressman, Krieger, and Strain together with a De Giorgi iteration. This stronger regularization rate illustrates the importance of the nonlinear nature of the diffusion in the analysis of the Landau equation and raises the question of determining whether this rate also happens for the Landau-Coulomb equation itself.

math.AP↗

arXiv2021

An optimal transportation principle for interacting paths and congestion

In this work we study a modification of the Monge-Kantorovich problem taking into account path dependence and interaction effects between particles. We prove existence of solutions under mild conditions on the data, and after imposing stronger conditions, we characterize the minimizers by relating them to an auxiliary Monge-Kantorovich problem of the more standard kind. With this notion of how particles interact and travel along paths, we produce a dual problem. The main novelty here is to incorporate an interaction effect to the optimal path transport problem. This covers for instance, $N$-body dynamics when the underlying measures are discrete. Lastly, our results include an extension of Brenier's theorem on optimal transport maps.

math.AP↗