  Characterising intractable high-dimensional random variables is one of the
fundamental challenges in stochastic computation. The recent surge of transport
maps offers a mathematical foundation and new insights for tackling this
challenge by coupling intractable random variables with tractable reference
random variables. This paper generalises the functional tensor-train
approximation of the inverse Rosenblatt transport recently developed by Dolgov
et al. (Stat Comput 30:603--625, 2020) to a wide class of high-dimensional
non-negative functions, such as unnormalised probability density functions.
First, we extend the inverse Rosenblatt transform to enable the transport to
general reference measures other than the uniform measure. We develop an
efficient procedure to compute this transport from a squared tensor-train
decomposition which preserves the monotonicity. More crucially, we integrate
the proposed order-preserving functional tensor-train transport into a nested
variable transformation framework inspired by the layered structure of deep
neural networks. The resulting deep inverse Rosenblatt transport significantly
expands the capability of tensor approximations and transport maps to random
variables with complicated nonlinear interactions and concentrated density
functions. We demonstrate the efficiency of the proposed approach on a range of
applications in statistical learning and uncertainty quantification, including
parameter estimation for dynamical systems and inverse problems constrained by
partial differential equations.
