Deep composition of tensor-trains using squared inverse Rosenblatt transports

TL;DR

This paper introduces a deep tensor-train based inverse Rosenblatt transport method for high-dimensional probability distributions, enabling efficient approximation and transformation of complex, nonlinear, and concentrated densities in statistical learning and uncertainty quantification.

Contribution

It generalizes the inverse Rosenblatt transform to broader measures, develops an efficient tensor-train computation, and integrates it into a deep layered framework for improved high-dimensional transport.

Findings

Efficient tensor-train based transport for complex densities

Enhanced capability for nonlinear high-dimensional transformations

Successful applications in dynamical systems and PDE-constrained inverse problems

Abstract

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.