Title: A theoretical analysis for inverse problems in optimal transport
Abstract: Estimating parameters from samples of an optimal probability distribution is crucial for applications ranging from socio-economic modeling to biological system analysis. In these cases, the probability distribution arises as the solution to an optimization problem that models how agents interact in static settings or how the system evolves over time in dynamic scenarios. In this work, we investigate the stability of a broad class of convex methods for estimating key parameters—typically a cost function for static problems and a potential function for dynamic problems. Our approach is based on unbalanced optimal transport (UOT) with entropic regularization. Unlike classical optimal transport, UOT allows for probability measures with varying total mass, making it well-suited for handling noisy or incomplete data and for modeling dynamical systems. I will show how this problem can be tackled using a convex loss function and discuss the associated recovery properties, including sample complexity bounds, as well as regularization properties. This is joint work with Francisco Andrade and Gabriel Peyré.