Sparse approximation of triangular transports. Part II: the infinite dimensional case

TL;DR

This paper studies the approximation of infinite-dimensional triangular transports between probability measures, demonstrating that rational functions can efficiently approximate these transports without the curse of dimensionality, with applications in high-dimensional Bayesian inference.

Contribution

It introduces a method to approximate infinite-dimensional triangular transports using rational functions, avoiding the curse of dimensionality in high-dimensional settings.

Findings

Rational function approximation of triangular transports is feasible in infinite dimensions.

The approach enables efficient sampling from high-dimensional measures.

Applications include Bayesian inference in infinite-dimensional Banach spaces.

Abstract

For two probability measures $\rho$ and $\pi$ on $[-1,1]^{\mathbb{N}}$ we investigate the approximation of the triangular Knothe-Rosenblatt transport $T:[-1,1]^{\mathbb{N}}\to [-1,1]^{\mathbb{N}}$ that pushes forward $\rho$ to $\pi$. Under suitable assumptions, we show that $T$ can be approximated by rational functions without suffering from the curse of dimension. Our results are applicable to posterior measures arising in certain inference problems where the unknown belongs to an (infinite dimensional) Banach space. In particular, we show that it is possible to efficiently approximately sample from certain high-dimensional measures by transforming a lower-dimensional latent variable.