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

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).(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

We propose a novel offline-online method that reduces the computational burden of posterior random variable characterization in statistical learning. This method learns the joint distribution of parameter and observable random variables in a tensor-train (TT) format during the offline phase. Then, in the online phase, it uses order-preserving conditional transport to describe the posterior random variables in real time based on newly observed data. This method relies on function approximation and includes thorough performance analysis, providing improved transport maps for high-dimensional observation and parameter problems.