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Session I.6 - Mathematical Foundations of Data Assimilation and Inverse Problems

Wednesday, June 14, 18:00 ~ 18:30

Statistical theory for transport-based generative modelling

Sven Wang

M.I.T., United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Measure transport provides a powerful toolbox for estimation and generative modelling of complicated probability distributions. The common principle is to learn a transport map between a simple reference distribution and a complicated target distribution. In this talk, we discuss recent advances in statistical guarantees for such methods. We discuss multiple relevant classes of maps: (1) triangular maps, which are the building blocks for 'autoregressive normalizing flows', (2) optimal transport maps and (3) ODE-based maps, where the coupling between reference and target is given by an ODE flow. This encompasses NeuralODEs, a popular method for generative modeling.

We derive non-asymptotic convergence rates for the distance between the transport-based estimator and the unknown 'ground truth' probability distribution, which converges to 0 algebraically in the statistical sample size. Our results imply that in certain cases, transport methods achieve minimax-optimal convergence rates for non-parametric density estimation, which was previously unknown.

Joint work with Youssef Marzouk (MIT, United States), Robert Ren (MIT, United States) and Jakob Zech (U Heidelberg, Germany).

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