# Bernoulli Flow Models: Self-Consistent Generative Modeling for Binary Data

arXiv: https://arxiv.org/abs/2610.11362 · HTML: https://mlsanity.com/paper/2610.11362/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

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

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning](https://mlsanity.com/paper/2208.04202/) (2022)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Progressive Distillation for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2202.00512/) (2022)
- [α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models](https://mlsanity.com/paper/2504.10283/) (2025)
- [Improving Vector-Quantized Image Modeling with Latent Consistency-Matching Diffusion](https://mlsanity.com/paper/2410.14758/) (2024)
- [Bit-Level Discrete Diffusion with Markov Probabilistic Models: An Improved Framework with Sharp Convergence Bounds under Minimal Assumptions](https://mlsanity.com/paper/2502.07939/) (2025)
- [Prediction--Loss Alignment for Sampler--Robust Flow Matching Training](https://mlsanity.com/paper/2602.10420/) (2026)
- [CoVAE: Consistency Training of Variational Autoencoders](https://mlsanity.com/paper/2507.09103/) (2025)
- [Binary Diffusion Probabilistic Model](https://mlsanity.com/paper/2501.13915/) (2025)

## Abstract (arXiv)

Binary diffusion models typically require a large number of function evaluations (NFEs) to generate high-quality samples, making practical inference computationally expensive. Reducing NFEs while preserving sample quality without distillation or additional training remains a significant challenge. Existing binary diffusion models define a discrete one-step forward path and then derive the reverse posterior. In low-NFE settings requiring cross-step sampling, they approximate the true multi-step likelihood with a single-step likelihood transition, which severely degrades sample quality. To address this fundamental limitation and decouple the generative dynamics from fixed discrete time steps, we propose Bernoulli Flow Models (BFM). Rather than relying on sequential one-step Markov diffusion chains, BFM defines a unified continuous global Bernoulli probability flow path between data distributions and pure noise, from which we derive analytical closed-form posterior transitions over arbitrary time intervals. Consequently, reducing the inference NFE is no longer an approximation based on skipping discrete steps; it only requires re-evaluating the analytical posterior over a new time grid. This eliminates the structural training-inference mismatch inherent to discrete chains and yields self-consistent low-NFE sampling. Experiments show that BFM is highly robust to aggressive NFE reduction. On LSUN Churches 256x256, a BFM trained with 256 steps achieves an FID of 9.22 using only 16 sampling steps, whereas the state-of-the-art discrete baseline degrades to 204.10. BFM also remains competitive with continuous and discrete generative baselines under standard full-step inference. These results establish BFM as a theoretically rigorous, self-consistent, and practically effective framework for fast binary data generation.