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

For two probability measures $\rho$ and $\pi$ with analytic densities on the $d$-dimensional cube $[-1,1]^d$, we investigate the approximation of the unique triangular monotone Knothe-Rosenblatt transport $T:[-1,1]^d\to [-1,1]^d$, such that 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 either sparse polynomial expansions or deep ReLU neural networks, such that the distance between $\tilde T\sharp\rho$ and $\pi$ decreases exponentially. More precisely, 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 the dimension of the ansatz space (or the size of the network) containing $\tilde T$; the notion of distance comprises the Hellinger distance, the total variation distance, the Wasserstein distance and the Kullback-Leibler divergence. Our construction guarantees $\tilde T$ to be a monotone triangular bijective transport on the hypercube $[-1,1]^d$. Analogous results hold for the inverse transport $S=T^{-1}$. The proofs are constructive, and we give an explicit a priori description of the ansatz space, which can be used for numerical implementations.