Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction

TL;DR

This paper introduces a scalable method using tensor-train representations and gradient-based dimension reduction to efficiently characterize posterior distributions in statistical learning, especially for high-dimensional problems.

Contribution

It develops a novel offline-online framework that leverages tensor-train formats for posterior characterization, extending transport maps with reordering and reparametrization heuristics.

Findings

Efficient posterior characterization in high-dimensional settings.

Enhanced transport map performance through variable reordering and layered compositions.

Validated on ODE and PDE-based statistical learning tasks.

Abstract

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).