Deep Composition of Tensor Trains using Squared Inverse Rosenblatt Transports

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 a recently developed functional tensor-train (FTT) approximation of the inverse Rosenblatt transport [14] to a wide class of high-dimensional nonnegative 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 FTT decomposition which preserves the monotonicity. More crucially, we integrate the proposed monotonicity-preserving FTT transport into a nested variable transformation framework inspired by 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 efficacy 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.