Branched transport is a variant of optimal transport where the cost is strictly sub-additive in the mass m to be transported (typically a concave power of m), thereby favoring grouped transport. As a result, the trajectories of the particles form a transport network with branching points, which may exhibit fractal-like features.This variational model was introduced to model various artificial or natural networks, such as communication and transport systems, plant root structures, and river networks.
In this talk, I will begin with an introduction to branched transport and its Lagrangian formalism, followed by an introduction to the problem of optimal quantization of measures. This problem consists in finding the best approximation of a measure by an atomic one with a fixed number of atoms, usually expressed via Wasserstein distances. Our aim is to replace these distances with the distances derived from branched transport. I will present some results on the asymptotics as the number of atoms tends to infinity, as well as on the uniformity properties of optimal quantizers.
This is a joint work with Mircea Petrache.