# Reverse Markov Learning: Multi-Step Generative Models for Complex Distributions

arXiv: https://arxiv.org/abs/2502.13747 · HTML: https://mlsanity.com/paper/2502.13747/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 22 references that are in the map.
- First posted on arXiv 19 Feb 2025.

## Builds on (most cited first, 7 of 7)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)

## Built on by (most cited first, 7 of 7)

- [EnScale: Temporally-consistent multivariate generative downscaling via proper scoring rules](https://mlsanity.com/paper/2509.26258/) (2025)
- [Generative diffusion posterior sampling for informative likelihoods](https://mlsanity.com/paper/2506.01083/) (2025)
- [Count Bridges enable Modeling and Deconvolving Transcriptomic Data](https://mlsanity.com/paper/2603.04730/) (2026)
- [Theoretical Analysis of Engression and Reverse Markov Engression](https://mlsanity.com/paper/2606.01002/) (2026)
- [Energy-Tweedie: Score meets Score, Energy meets Energy](https://mlsanity.com/paper/2512.23818/) (2025)
- [Improved Distributional Diffusion Models](https://mlsanity.com/paper/2609.37147/) (2026)
- [Extrapolation Guarantees for Perturbation Modeling Under the Additive Latent Shift Assumption](https://mlsanity.com/paper/2504.18522/) (2025)

## Abstract (arXiv)

Learning complex distributions is a fundamental challenge in contemporary applications. Shen and Meinshausen (2024) introduced engression, a generative approach based on scoring rules that maps noise (and covariates, if available) directly to data. While effective, engression can struggle with highly complex distributions, such as those encountered in image data. In this work, we propose reverse Markov learning (RML), a framework that defines a general forward process transitioning from the target distribution to a known distribution (e.g., Gaussian) and then learns a reverse Markov process using multiple engression models. This reverse process reconstructs the target distribution step by step. This framework accommodates general forward processes, allows for dimension reduction, and naturally discretizes the generative process. In the special case of diffusion-based forward processes, RML provides an efficient discretization strategy for both training and inference in diffusion models. We further introduce an alternating sampling scheme to enhance post-training performance. Our statistical analysis establishes error bounds for RML and elucidates its advantages in estimation efficiency and flexibility in forward process design. Empirical results on simulated and climate data corroborate the theoretical findings, demonstrating the effectiveness of RML in capturing complex distributions.