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

TL;DR

This paper demonstrates that for probability measures with analytic densities, the triangular transport map can be approximated efficiently using sparse polynomials or neural networks, achieving exponential error decay in high dimensions.

Contribution

It introduces explicit constructions of sparse polynomial and neural network approximations for the triangular transport, with proven exponential error bounds in various distances.

Findings

Exponential decay of approximation error with respect to ansatz size.

Constructive proofs providing explicit approximation schemes.

Guarantees monotonicity and bijectivity of the approximations.

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 $T:[-1,1]^d\to [-1,1]^d$, 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.