The Transport Map Computed by Iterated Function System

The transport map in the optimal transport model plays an important role in the machine learning and statistics fields, and the approximation of the transport map is significant for application. Since the transport map has no explicit expression in the general case, representations of such a map or realizing its action are often intractable as the dimension increases. In this paper, we adopt a new perspective to approximate the transport map by using an iterated function system: the transport map is constructed through a composition of some iterated maps. The source measure in the optimal transport model is transferred through the iterated maps, and the corresponding sequence of iterated measures has been produced. We show that there is an \(\epsilon \) $ϵ$ -optimal transport map between the iterated measure and target measure in each iteration of the iterated function system. Theoretical convergence analysis of the sequence of iterated measures is performed, the iterated measures converge to the target measure in the optimal transport model, and we get a geometric convergence rate if the iterated maps are Lipschitz continuous and independent from each other. Finally, we give the statistical estimation error of the transport map approximated by the iterated function system.