  For two probability measures $\rho$ and $\pi$ on $[-1,1]^{\mathbb{N}}$ we
investigate the approximation of the triangular Knothe-Rosenblatt transport
$T:[-1,1]^{\mathbb{N}}\to [-1,1]^{\mathbb{N}}$ that pushes forward $\rho$ to
$\pi$. Under suitable assumptions, we show that $T$ can be approximated by
rational functions without suffering from the curse of dimension. Our results
are applicable to posterior measures arising in certain inference problems
where the unknown belongs to an (infinite dimensional) Banach space. In
particular, we show that it is possible to efficiently approximately sample
from certain high-dimensional measures by transforming a lower-dimensional
latent variable.
