# Theoretical Analysis of Engression and Reverse Markov Engression

arXiv: https://arxiv.org/abs/2606.01002 · HTML: https://mlsanity.com/paper/2606.01002/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 23 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Brain Imaging Generation with Latent Diffusion Models](https://mlsanity.com/paper/2209.07162/) (2022)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [CARD: Classification and Regression Diffusion Models](https://mlsanity.com/paper/2206.07275/) (2022)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)
- [Distributional Diffusion Models with Scoring Rules](https://mlsanity.com/paper/2502.02483/) (2025)
- [Reliable Generation of Privacy-preserving Synthetic Electronic Health Record Time Series via Diffusion Models](https://mlsanity.com/paper/2310.15290/) (2023)
- [Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networks](https://mlsanity.com/paper/2002.03938/) (2020)
- [Wasserstein Generative Regression](https://mlsanity.com/paper/2306.15163/) (2023)
- [Geometrical Insights for Implicit Generative Modeling](https://mlsanity.com/paper/1712.07822/) (2017)
- [Reverse Markov Learning: Multi-Step Generative Models for Complex Distributions](https://mlsanity.com/paper/2502.13747/) (2025)
- [Frugal, Flexible, Faithful: Causal Data Simulation via Frengression](https://mlsanity.com/paper/2508.01018/) (2025)

## Built on by (most cited first, 4 of 4)

- [Error Analysis of Neural-Network-Based Engression](https://mlsanity.com/paper/2607.27723/) (2026)
- [Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression](https://mlsanity.com/paper/2609.23789/) (2026)
- [Sufficiently Reduced Distributional Regression](https://mlsanity.com/paper/2609.29291/) (2026)
- [Generalized Engression Models](https://mlsanity.com/paper/2610.01823/) (2026)

## Abstract (arXiv)

Engression is a recently proposed and effective framework for conditional distribution learning. Its multi-step Reverse Markov extension further improves generative flexibility by decomposing complex conditional sampling into sequential reverse transitions. Despite their strong empirical performance, rigorous finite-sample statistical guarantees for these methods remain unavailable. In this paper, under deep neural network parameterizations, we establish nonasymptotic convergence bounds for Engression by directly controlling the Energy Distance between the learned and target conditional distributions. For the Reverse Markov framework, we further develop an Energy-Distance-based chain rule that enables a rigorous analysis of error propagation across reverse steps. Our analysis yields corresponding excess-risk bounds that are near-optimal up to logarithmic factors relative to the classical minimax rate over a general H"older class.