# ReSFlow: Rectifying Categorical Flows on Statistical Manifolds for One-Step Generation

arXiv: https://arxiv.org/abs/2610.13225 · HTML: https://mlsanity.com/paper/2610.13225/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 28 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 20 of 20)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Energy-Based Diffusion Language Models for Text Generation](https://mlsanity.com/paper/2410.21357/) (2024)
- [Beyond Autoregression: Fast LLMs via Self-Distillation Through Time](https://mlsanity.com/paper/2410.21035/) (2024)
- [Dirichlet Diffusion Score Model for Biological Sequence Generation](https://mlsanity.com/paper/2305.10699/) (2023)
- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective](https://mlsanity.com/paper/2412.03487/) (2024)
- [Discrete Copula Diffusion](https://mlsanity.com/paper/2410.01949/) (2024)
- [Distillation of Discrete Diffusion through Dimensional Correlations](https://mlsanity.com/paper/2410.08709/) (2024)
- [Categorical Flow Matching on Statistical Manifolds](https://mlsanity.com/paper/2405.16441/) (2024)
- [ReDi: Rectified Discrete Flow](https://mlsanity.com/paper/2507.15897/) (2025)
- [Riemannian Variational Flow Matching for Material and Protein Design](https://mlsanity.com/paper/2502.12981/) (2025)
- [Riemannian Consistency Model](https://mlsanity.com/paper/2510.00983/) (2025)

## Abstract (arXiv)

Generative modeling for discrete data has been significantly advanced by learning continuous vector fields on the statistical manifolds to transport simple priors to target discrete distributions. However, because these continuous-time flows are not constrained to follow geodesics on the statistical manifold, thereby requiring multi-step numerical integration for simulation and leading to substantial computational costs during inference. In this work, drawing inspiration from the "straight and fast" properties of Rectified Flow in continuous Euclidean space, we propose the Rectified Statistical Flow (ReSFlow) model, which enables one-step categorical generation on statistical manifolds by explicitly learning geodesic trajectories that adhere to the intrinsic Riemannian geometry. By rectifying the flow on the statistical manifold, we theoretically guarantee that trajectories between prior and target distributions approximate geodesics. This rectification significantly reduces the transport cost by aligning the learned trajectories directly along geodesics on the statistical manifold, allowing the model to map pure noise to target discrete data in a single step. Through extensive experiments on diverse real-world generative tasks, ranging from image and text to biological domains, we demonstrate ReSFlow's superior one-step generation quality, which truly embodies geodesically optimal and rapid categorical generation. Our code is available at https://github.com/xiaohuanjia/ReSFlow.