Deep Composition of Tensor Trains using Squared Inverse Rosenblatt Transports

Tiangang Cui, Sergey Dolgov

Research output: Contribution to journalArticle

8 Citations (SciVal)


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.

Original languageEnglish
Number of pages47
JournalFoundations of Computational Mathematics
Publication statusPublished - 21 Sept 2021

Bibliographical note

Funding Information:
The author would like to thank Y. Marzouk and R. Scheichl for many insightful discussions. T. Cui acknowledges support from the Australian Research Council, under grant number CE140100049. S. Dolgov acknowledges support from the International Visitor Program of Sydney Mathematical Research Institute, and from the EPSRC New Investigator Award EP/T031255/1.

Publisher Copyright:
© 2021, The Author(s).


  • Deep transport maps
  • Inverse problems
  • Rosenblatt transport
  • Tensor-train
  • Uncertainty quantification

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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