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

Abstract

AbstractFor two probability measures $${\rho }$$ ρ and $${\pi }$$ π with analytic densities on the d-dimensional cube $$[-1,1]^d$$ [ - 1 , 1 ] d , we investigate the approximation of the unique triangular monotone Knothe–Rosenblatt transport $$T:[-1,1]^d\rightarrow [-1,1]^d$$ T : [ - 1 , 1 ] d → [ - 1 , 1 ] d , such that the pushforward $$T_\sharp {\rho }$$ T ♯ ρ equals $${\pi }$$ π . It is shown that for $$d\in {{\mathbb {N}}}$$ d ∈ N there exist approximations $${\tilde{T}}$$ T ~ of T, based on either sparse polynomial expansions or deep ReLU neural networks, such that the distance between $${\tilde{T}}_\sharp {\rho }$$ T ~ ♯ ρ and $${\pi }$$ π decreases exponentially. More precisely, we prove error bounds of the type $$\exp (-\beta N^{1/d})$$ exp ( - β N 1 / d ) (or $$\exp (-\beta N^{1/(d+1)})$$ exp ( - β N 1 / ( d + 1 ) ) for neural networks), where N refers to the dimension of the ansatz space (or the size of the network) containing $${\tilde{T}}$$ T ~ ; the notion of distance comprises the Hellinger distance, the total variation distance, the Wasserstein distance and the Kullback–Leibler divergence. Our construction guarantees $${\tilde{T}}$$ T ~ to be a monotone triangular bijective transport on the hypercube $$[-1,1]^d$$ [ - 1 , 1 ] d . Analogous results hold for the inverse transport $$S=T^{-1}$$ 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.