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For two probability measures $\rho$ and $\pi$ with analytic densities on the$d$-dimensional cube $[-1,1]^d$, we investigate the approximation of the uniquetriangular monotone Knothe-Rosenblatt transport $T:[-1,1]^d\to [-1,1]^d$, suchthat the pushforward $T_\sharp\rho$ equals $\pi$. It is shown that for$d\in\mathbb{N}$ there exist approximations $\tilde T$ of $T$, based on eithersparse polynomial expansions or deep ReLU neural networks, such that thedistance between $\tilde T_\sharp\rho$ and $\pi$ decreases exponentially. Moreprecisely, we prove error bounds of the type $\exp(-\beta N^{1/d})$ (or$\exp(-\beta N^{1/(d+1)})$ for neural networks), where $N$ refers to thedimension of the ansatz space (or the size of the network) containing $\tildeT$; the notion of distance comprises the Hellinger distance, the totalvariation distance, the Wasserstein distance and the Kullback-Leiblerdivergence. Our construction guarantees $\tilde T$ to be a monotone triangularbijective transport on the hypercube $[-1,1]^d$. Analogous results hold for theinverse transport $S=T^{-1}$. The proofs are constructive, and we give anexplicit a priori description of the ansatz space, which can be used fornumerical implementations.

Sparse approximation of triangular transports. Part I: the finite dimensional case