# Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates

arXiv: https://arxiv.org/abs/2112.15595 · HTML: https://mlsanity.com/paper/2112.15595/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 43 references that are in the map.
- First posted on arXiv 31 Dec 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [An introduction to sampling via measure transport](https://mlsanity.com/paper/1602.05023/) (2016)

## Built on by (most cited first, 16 of 16)

- [Indeterminacy in Generative Models: Characterization and Strong Identifiability](https://mlsanity.com/paper/2206.00801/) (2022)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [A Fixed-Point Approach for Causal Generative Modeling](https://mlsanity.com/paper/2404.06969/) (2024)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Expected information gain estimation via density approximations: Sample allocation and dimension reduction](https://mlsanity.com/paper/2411.08390/) (2024)
- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026)
- [Convergence Rates for Distribution Matching with Sliced Optimal Transport](https://mlsanity.com/paper/2602.10691/) (2026)
- [Causal Order Discovery based on Monotonic SCMs](https://mlsanity.com/paper/2410.19870/) (2024)
- [On Measuring Intrinsic Causal Attributions in Deep Neural Networks](https://mlsanity.com/paper/2505.09660/) (2025)
- [FlowSDR: Sufficient Dimension Reduction via Conditional Normalizing Flows](https://mlsanity.com/paper/2606.01346/) (2026)
- [Learning the Structure of Triangular Transport Maps](https://mlsanity.com/paper/2609.37122/) (2026)
- [Multifidelity Formulations for Triangular Transport](https://mlsanity.com/paper/2610.00698/) (2026)

## Abstract (arXiv)

Triangular flows, also known as Kn"othe-Rosenblatt measure couplings, comprise an important building block of normalizing flow models for generative modeling and density estimation, including popular autoregressive flow models such as real-valued non-volume preserving transformation models (Real NVP). We present statistical guarantees and sample complexity bounds for triangular flow statistical models. In particular, we establish the statistical consistency and the finite sample convergence rates of the Kullback-Leibler estimator of the Kn"othe-Rosenblatt measure coupling using tools from empirical process theory. Our results highlight the anisotropic geometry of function classes at play in triangular flows, shed light on optimal coordinate ordering, and lead to statistical guarantees for Jacobian flows. We conduct numerical experiments on synthetic data to illustrate the practical implications of our theoretical findings.