  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).
