Hossein Gorji

When Samples Correlate Symmetrically: Particle Dynamics for Optimal Transport

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Abstract 

In this talk, I will introduce orthogonal coupling dynamics (OCD), a dynamical approach to address the Monge–Kantorovich problem. The main idea is to evolve a coupling between two fixed marginals by projecting the gradient descent of the transport cost onto a marginal-preserving tangent subspace. The resulting dynamics is expressed through conditional expectations: it keeps each marginal unchanged while monotonically reducing the expected transport cost.

For the quadratic cost, OCD admits a simple particle interpretation in which samples interact through regression-like conditional averages. This viewpoint connects optimal transport to kinetic and opinion-dynamics models, and leads to a nonparametric Monte Carlo algorithm for estimating transport maps from samples. I will present the main structural properties of the dynamics, including marginal preservation, cost decay, instability of suboptimal couplings, a Vlasov-type formulation, and a sharp-descent variational characterization. I will also discuss the Gaussian case, where the cross-correlation follows a Riccati equation and, for commuting covariance matrices, converges exponentially to the optimal transport coupling. The talk will conclude with numerical examples, open questions, and possible extensions toward multi-marginal optimal transport.

Presenters

Prof. Hossein Gorji

Brief Biography

Hossein Gorji is a Staff Scientist at Laboratory of Computational Engineering at Empa, Lecturer of Applied Mathematics at EPFL, and Adjunct Research Assistant Professor at the University of Southern California. He received his Ph.D. in Mechanical Engineering from ETH Zürich in 2014, where his dissertation on Fokker–Planck algorithms for rarefied gas flows was awarded the ETH Medal. His research lies at the interface of kinetic theory, particle methods, and machine learning for scientific computing, with current interests including optimal transport, distribution learning, robust forecasting, Fokker–Planck and Boltzmann-type kinetic models, and epidemic modeling. He has held research and visiting positions at ETH Zürich, RWTH Aachen, EPFL, Caltech, Stanford, TU Munich, and Institut Polytechnique de Bordeaux. His work has been supported by the Swiss National Science Foundation, the German Research Foundation, the Swiss Federal Office of Public Health, and competitive fellowships, including the SNSF Ambizione Award.