Sven Wang (Humboldt University Berlin)

Statistical convergence rates for transport- and ODE-based generative models

Abstract. Measure transport provides a powerful toolbox for estimation and generative modelling of complicated probability distributions. The common principle is to learn a transport map which couples a tractable (e.g. uniform or normal) reference distribution to some complicated target distribution, e.g. by maximizing a likelihood objective. In this talk, we discuss recent advances in statistical convergence guarantees for such methods. While a general theory is developed, we will primarily treat (1) triangular maps which are the building blocks for ’autoregressive normalizing flows’ and (2) ODE-based maps, defined through an ODE flow. The latter encompasses NeuralODEs, a popular method for generative modeling. Our results imply that transport methods achieve minimax-optimal convergence rates for non-parametric density estimation over Hölder classes on the unit cube.

Joint with Youssef Marzouk, Robert Ren and Jakob Zech.