Sparse Approximation of Triangular Transports, Part I: The Finite-Dimensional Case

Abstract

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]^drightarrow [-1,1]^d, such that the pushforward T_sharp {rho } equals {pi }. It is shown that for din {{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.