  We present a novel offline-online method to mitigate the computational burden
of the characterization of posterior random variables in statistical learning.
In the offline phase, the proposed method learns the joint law of the parameter
random variables and the observable random variables in the tensor-train (TT)
format. In the online phase, the resulting order-preserving conditional
transport can characterize the posterior random variables given newly observed
data in real time. Compared with the state-of-the-art normalizing flow
techniques, the proposed method relies on function approximation and is
equipped with a thorough performance analysis. The function approximation
perspective also allows us to further extend the capability of transport maps
in challenging problems with high-dimensional observations and high-dimensional
parameters. On the one hand, we present novel heuristics to reorder and/or
reparametrize the variables to enhance the approximation power of TT. On the
other hand, we integrate the TT-based transport maps and the parameter
reordering/reparametrization into layered compositions to further improve the
performance of the resulting transport maps. We demonstrate the efficiency of
the proposed method on various statistical learning tasks in ordinary
differential equations (ODEs) and partial differential equations (PDEs).
