Sparse approximation of triangular transports on bounded domains
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Let ρ and π be two probability measures on [-1,1]^d with positive and analytic Lebesgue densities. We investigate the approximation of the unique triangular monotone (Knothe-Rosenblatt) transport T:[-1,1]^d→ [-1,1]^d, such that the pushforward T_♯ρ equals π. It is shown that for d∈ℕ there exist approximations T̃ of T based on either sparse polynomial expansions or ReLU networks, such that the distance between T̃_♯ρ and π decreases exponentially. More precisely, we show error bounds of the type (-β N^1/d) (or (-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 T̃; the notion of distance comprises, among others, the Hellinger distance and the Kullback–Leibler divergence. The construction guarantees T̃ 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. Additionally we discuss the high-dimensional case: for d=∞ a dimension-independent algebraic convergence rate is proved for a class of probability measures occurring widely in Bayesian inference for uncertainty quantification, thus verifying that the curse of dimensionality can be overcome in this setting.
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