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 transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling with tractable reference variables. This paper generalises functional tensor-train approximation inverse Rosenblatt recently developed Dolgov et al. (Stat Comput 30:603--625, 2020) to wide class non-negative functions, such as unnormalised probability density functions. First, we extend transform enable general measures other than uniform measure. We develop an efficient procedure compute from squared decomposition which preserves monotonicity. More crucially, integrate proposed order-preserving into nested variable transformation framework inspired layered structure deep neural networks. resulting significantly expands capability tensor approximations complicated nonlinear interactions concentrated demonstrate efficiency approach on range applications statistical learning uncertainty quantification, including parameter estimation dynamical systems problems constrained partial differential equations.