Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Sparse approximation of triangular transports on bounded domains

Prof. Dr. Jakob Zech talk

**Date:** 21.10.20 **Time:** 16.15 - 17.45 **Room:** Online ZHACM

Transport maps coupling two different measures can be used to sample from arbitrarily complex distributions. One of the main applications of this approach concerns Bayesian inference, where sampling from a posterior distribution facilitates making predictions based on partial and noisy measurments. In this talk we investigate the approximation of triangular transports $T:[-1,1]^d\to [-1,1]^d$ on the $d$-dimensional unit cube by polynomial expansions and ReLU networks. Specifically, given a reference and a target probability measure with positive and analytic Lebesgue densities on $[-1,1]^d$, we show that the unique Knothe-Rosenblatt transport, which pushes forward the reference to the target, can be approximated at an exponential rate in case $d<\infty$. These results are generalized to $d=\infty$, within a setting which incorporates many posterior densities occurring in PDE-driven Bayesian inverse problems. In the infinite dimensional case ($d=\infty$) we verify an algebraic convergence rate, which shows that the curse of dimensionality can be overcome.