离散空间中实现一步生成,无需迭代或积分。
Discrete MeanFlow: One-Step Generation via Conditional Transition Kernels

- 用条件转移核替代连续轨迹,直接建模概率质量转移
- 学习到的转移核可精确恢复真实马尔可夫链,误差极小
- 适合需要快速生成的离散序列任务,如文本、代码
MeanFlow 在连续空间中通过学习时间区间的平均速度实现一步生成,但离散状态空间无光滑轨迹和空间导数,无法直接应用。本文提出 Discrete MeanFlow,将点的运动替换为有限状态间概率质量的传输。核心对象是连续时间马尔可夫链(CTMC)的条件转移核,从中定义平均转移速率以衡量时间区间内概率变化的均值。我们证明了 Discrete MeanFlow 恒等式,该恒等式将有限区间速率与终点处的瞬时 CTMC 生成器关联,柯尔莫哥洛夫前向方程取代了连续 MeanFlow 中的空间链式法则。基于此,我们采用边界构造设计参数化转移核,确保输出为有效概率分布且精确满足边界条件,无需辅助损失。由于学习到的核本身即为概率分布,生成仅需一次前向传播加一次类别采样,无需迭代去噪、常微分方程积分或多步优化。我们在精确有限状态马尔可夫链上验证框架,学习核以高精度恢复解析真值;在不同词表大小和序列长度的因子化合成序列生成任务上也表现优异。
原文摘要 · Abstract (English)
MeanFlow enables one-step generation in continuous spaces by learning an average velocity over a time interval rather than the instantaneous velocity field of flow matching. However, discrete state spaces do not have smooth trajectories or spatial derivatives, so the continuous formulation does not directly apply. We introduce Discrete MeanFlow, which replaces the motion of a point with the transport of probability mass over finite states. Our key object is the conditional transition kernel of a continuous-time Markov chain (CTMC), from which we define a mean discrete rate that measures the average change in transition probability over a time interval. We prove a Discrete MeanFlow identity that relates this finite-interval rate to the instantaneous CTMC generator at the endpoint, with the Kolmogorov forward equation replacing the spatial chain rule of continuous MeanFlow. Based on this identity, we parameterize the transition kernel directly using a boundary-by-construction design that guarantees valid probability outputs and exact boundary conditions without auxiliary losses. Since the learned kernel is itself a probability distribution, generation reduces to a single forward pass followed by one categorical draw meaning no iterative denoising, ODE integration, or multi-step refinement is required. We validate the framework on exact finite-state Markov chains, where the learned kernel recovers the analytical ground truth to high precision, and on factorized synthetic sequence generation tasks with varying alphabet sizes and sequence lengths.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品;中文卡片由大模型生成,请以原文为准。