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

2020/06/12 by Zech, Jakob, Marzouk, Youssef · 1 citation
#32D05 #41A10 #41A25 #41A46 #62D99 #65D15 #FOS: Mathematics #Numerical Analysis (math.NA) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2006.06994

Abstract

For two probability measures ρ and π 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→ [-1,1]^d, such that the pushforward T_\sharpρ equals π. It is shown that for d∈ℕ there exist approximations T of T, based on either sparse polynomial expansions or deep ReLU neural networks, such that the distance between T_\sharpρ and π decreases exponentially. More precisely, we prove error bounds of the type exp(-βN^1/d) (or exp(-βN^1/(d+1)) for neural networks), where N refers to the dimension of the ansatz space (or the size of the network) containing T; the notion of distance comprises the Hellinger distance, the total variation distance, the Wasserstein distance and the Kullback-Leibler divergence. Our construction guarantees 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.

Cited by

• Estimating systematic errors in Bayesian inversion using transport maps

Related

• Sparse approximation of triangular transports. Part II: the infinite dimensional case
• Efficient estimates of optimal transport via low-dimensional embeddings
• A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions
• Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps
• Distribution learning via neural differential equations: minimal energy regularization and approximation theory