# On the representation and learning of monotone triangular transport maps

arXiv: https://arxiv.org/abs/2009.10303 · HTML: https://mlsanity.com/paper/2009.10303/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2020.

## Built on by (most cited first, 30 of 41)

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- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
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- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Learning non-Gaussian graphical models via Hessian scores and triangular transport](https://mlsanity.com/paper/2101.03093/) (2021)
- [Variational Bayesian Optimal Experimental Design with Normalizing Flows](https://mlsanity.com/paper/2404.13056/) (2024)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Ensemble transport smoothing. Part II: Nonlinear updates](https://mlsanity.com/paper/2210.17435/) (2022)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
- [A generative flow for conditional sampling via optimal transport](https://mlsanity.com/paper/2307.04102/) (2023)
- [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024)
- [Measure transfer via stochastic slicing and matching](https://mlsanity.com/paper/2307.05705/) (2023)
- [Ensemble transport smoothing. Part I: Unified framework](https://mlsanity.com/paper/2210.17000/) (2022)
- [Expected information gain estimation via density approximations: Sample allocation and dimension reduction](https://mlsanity.com/paper/2411.08390/) (2024)
- [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025)
- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Self-reinforced polynomial approximation methods for concentrated probability densities](https://mlsanity.com/paper/2303.02554/) (2023)
- [A transport approach to sequential simulation-based inference](https://mlsanity.com/paper/2308.13940/) (2023)
- [Density Estimation via Measure Transport: Outlook for Applications in the Biological Sciences](https://mlsanity.com/paper/2309.15366/) (2023)
- [Optimal Transportation by Orthogonal Coupling Dynamics](https://mlsanity.com/paper/2410.08060/) (2024)
- [Learning local neighborhoods of non-Gaussian graphical models: A measure transport approach](https://mlsanity.com/paper/2503.13899/) (2025)
- [Neural Triangular Transport Maps: A New Approach Towards Sampling in Lattice QCD](https://mlsanity.com/paper/2510.13112/) (2025)
- [Incremental Non-Gaussian Inference for SLAM Using Normalizing Flows](https://mlsanity.com/paper/2110.00876/) (2021)
- [Deep importance sampling using tensor trains with application to a priori and a posteriori rare event estimation](https://mlsanity.com/paper/2209.01941/) (2022)
- [Transport map unadjusted Langevin algorithms: learning and discretizing perturbed samplers](https://mlsanity.com/paper/2302.07227/) (2023)
- [Sequential transport maps using SoS density estimation and α-divergences](https://mlsanity.com/paper/2402.17943/) (2024)
- [Adaptive Nonlinear Data Assimilation through P-Spline Triangular Measure Transport](https://mlsanity.com/paper/2603.19058/) (2026)
- [Copula Active Subspaces I: A Score-Covariance Method for Reduced-Order Non-Gaussian Density Estimation](https://mlsanity.com/paper/2609.36142/) (2026)

## Abstract (arXiv)

Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport mapsunicodex2014approximations of the Knotheunicodex2013Rosenblatt (KR) rearrangementunicodex2014are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.