# Self-reinforced polynomial approximation methods for concentrated probability densities

arXiv: https://arxiv.org/abs/2303.02554 · HTML: https://mlsanity.com/paper/2303.02554/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 56 references that are in the map.
- First posted on arXiv 5 Mar 2023.

## Builds on (most cited first, 12 of 12)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations](https://mlsanity.com/paper/1402.1694/) (2014)
- [Variational Inference with Continuously-Indexed Normalizing Flows](https://mlsanity.com/paper/2007.05426/) (2020)
- [Deep composition of tensor-trains using squared inverse Rosenblatt transports](https://mlsanity.com/paper/2007.06968/) (2020)
- [Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction](https://mlsanity.com/paper/2106.04170/) (2021)
- [Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems](https://mlsanity.com/paper/2102.13245/) (2021)
- [Sequential Ensemble Transform for Bayesian Inverse Problems](https://mlsanity.com/paper/1909.09591/) (2019)

## Built on by (most cited first, 2 of 2)

- [Sequential transport maps using SoS density estimation and α-divergences](https://mlsanity.com/paper/2402.17943/) (2024)
- [DPOT: A DeepParticle method for Computation of Optimal Transport with convergence guarantee](https://mlsanity.com/paper/2506.23429/) (2025)

## Abstract (arXiv)

Transport map methods offer a powerful statistical learning tool that can couple a target high-dimensional random variable with some reference random variable using invertible transformations. This paper presents new computational techniques for building the Knothe--Rosenblatt (KR) rearrangement based on general separable functions. We first introduce a new construction of the KR rearrangement -- with guaranteed invertibility in its numerical implementation -- based on approximating the density of the target random variable using tensor-product spectral polynomials and downward closed sparse index sets. Compared to other constructions of KR arrangements based on either multi-linear approximations or nonlinear optimizations, our new construction only relies on a weighted least square approximation procedure. Then, inspired by the recently developed deep tensor trains (Cui and Dolgov, Found. Comput. Math. 22:1863--1922, 2022), we enhance the approximation power of sparse polynomials by preconditioning the density approximation problem using compositions of maps. This is particularly suitable for high-dimensional and concentrated probability densities commonly seen in many applications. We approximate the complicated target density by a composition of self-reinforced KR rearrangements, in which previously constructed KR rearrangements -- based on the same approximation ansatz -- are used to precondition the density approximation problem for building each new KR rearrangement. We demonstrate the efficiency of our proposed methods and the importance of using the composite map on several inverse problems governed by ordinary differential equations (ODEs) and partial differential equations (PDEs).