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

arXiv: https://arxiv.org/abs/2106.04170 · HTML: https://mlsanity.com/paper/2106.04170/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 65 references that are in the map.
- First posted on arXiv 8 Jun 2021.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems](https://mlsanity.com/paper/1403.4290/) (2014)
- [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013)
- [Variational Inference with Continuously-Indexed Normalizing Flows](https://mlsanity.com/paper/2007.05426/) (2020)
- [Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction](https://mlsanity.com/paper/1510.06053/) (2015)
- [Deep composition of tensor-trains using squared inverse Rosenblatt transports](https://mlsanity.com/paper/2007.06968/) (2020)

## Built on by (most cited first, 9 of 9)

- [Learning to solve Bayesian inverse problems: An amortized variational inference approach using Gaussian and Flow guides](https://mlsanity.com/paper/2305.20004/) (2023)
- [Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities](https://mlsanity.com/paper/2406.13036/) (2024)
- [Self-reinforced polynomial approximation methods for concentrated probability densities](https://mlsanity.com/paper/2303.02554/) (2023)
- [Dimension reduction via score ratio matching](https://mlsanity.com/paper/2410.19990/) (2024)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Deep importance sampling using tensor trains with application to a priori and a posteriori rare event estimation](https://mlsanity.com/paper/2209.01941/) (2022)
- [Sequential transport maps using SoS density estimation and α-divergences](https://mlsanity.com/paper/2402.17943/) (2024)
- [Goal-oriented learning of stochastic differential equations using error bounds on path-space observables](https://mlsanity.com/paper/2603.20467/) (2026)
- [Gradient-free Riemannian Langevin Sampler](https://mlsanity.com/paper/2607.07519/) (2026)

## Abstract (arXiv)

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).