Self-reinforced polynomial approximation methods for concentrated probability densities

TL;DR

This paper introduces novel computational techniques for constructing the Knothe--Rosenblatt rearrangement using spectral polynomials and compositions of maps, improving approximation of high-dimensional, concentrated probability densities.

Contribution

It presents a new invertible construction of the KR rearrangement based on spectral polynomial approximation and enhances its power through composition of maps for high-dimensional densities.

Findings

Efficient approximation of complex target densities using self-reinforced KR rearrangements.

Improved handling of high-dimensional, concentrated probability densities.

Demonstrated effectiveness on inverse problems governed by ODEs and PDEs.

Abstract

Transport map methods offer a powerful statistical learning tool that can couple a target high-dimensional random variable with some reference random variable using invertible transformations. This paper presents new computational techniques for building the Knothe--Rosenblatt (KR) rearrangement based on general separable functions. We first introduce a new construction of the KR rearrangement -- with guaranteed invertibility in its numerical implementation -- based on approximating the density of the target random variable using tensor-product spectral polynomials and downward closed sparse index sets. Compared to other constructions of KR arrangements based on either multi-linear approximations or nonlinear optimizations, our new construction only relies on a weighted least square approximation procedure. Then, inspired by the recently developed deep tensor trains (Cui and Dolgov, Found. Comput. Math. 22:1863--1922, 2022), we enhance the approximation power of sparse polynomials by preconditioning the density approximation problem using compositions of maps. This is particularly suitable for high-dimensional and concentrated probability densities commonly seen in many applications. We approximate the complicated target density by a composition of self-reinforced KR rearrangements, in which previously constructed KR rearrangements -- based on the same approximation ansatz -- are used to precondition the density approximation problem for building each new KR rearrangement. We demonstrate the efficiency of our proposed methods and the importance of using the composite map on several inverse problems governed by ordinary differential equations (ODEs) and partial differential equations (PDEs).