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

Binary diffusion models typically require a massive number of function evaluations (NFEs) to generate high-quality samples, making practical inference computationally expensive. Reducing NFEs while preserving sample quality without relying on distillation or additional training remains a significant challenge. However, existing binary diffusion models sequentially define a discrete one-step forward path and subsequently derive the reverse posterior. In low-NFE scenarios that require cross-step sampling, these models incorrectly approximate the true multi-step likelihood using a single-step likelihood formulation, which severely degrades sample quality. To address this fundamental limitation and completely decouple the generative dynamics from fixed discrete time steps, we propose Bernoulli Flow Models (BFM). Rather than building upon sequential one-step Markov diffusion chains, we predefine 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 in BFM is no longer an approximation of skipped discrete steps; it simply requires re-evaluating the analytical posterior on the new time intervals. This eliminates the structural training-inference mismatch inherent to discrete chains, yielding strictly self-consistent low-NFE sampling. Experimental results show that BFM is highly robust to aggressive NFE reduction. On the LSUN Churches 256x256 dataset, a 256-step-trained BFM achieves an FID of 9.31 when sampled with only 16 steps, whereas the state-of-the-art discrete baseline severely degrades to 204.10. BFM also remains competitive with both continuous and discrete generative baselines under standard full-step inference. Ultimately, these results establish BFM as a theoretically rigorous, self-consistent, and practically effective framework for fast binary data generation.