Flow matching has emerged as a powerful framework for generative modeling, with recent empirical successes highlighting the effectiveness of signal-space prediction ($x$-prediction). In this work, we investigate the transfer of this paradigm to binary manifolds, a fundamental setting for generative modeling of discrete data. While $x$-prediction remains effective, we identify a latent structural mismatch that arises when it is coupled with velocity-based objectives ($v$-loss), leading to a time-dependent singular weighting that amplifies gradient sensitivity to approximation errors. Motivated by this observation, we formalize prediction-loss alignment as a necessary condition for flow matching training. We prove that re-aligning the objective to the signal space ($x$-loss) eliminates the singular weighting, yielding uniformly bounded gradients and enabling robust training under uniform timestep sampling without reliance on heuristic schedules. Finally, with alignment secured, we examine design choices specific to binary data, revealing a topology-dependent distinction between probabilistic objectives (e.g., cross-entropy) and geometric losses (e.g., mean squared error). Together, these results provide theoretical foundations and practical guidelines for robust flow matching on binary -- and related discrete -- domains, positioning signal-space alignment as a key principle for robust diffusion learning.

翻译：流匹配已成为生成建模的强大框架，其近期实证成果突显了信号空间预测（$x$-预测）的有效性。本文研究将该范式迁移至二元流形——离散数据生成建模的一个基础设定。虽然$x$-预测依然有效，但我们发现当其与基于速度的目标函数（$v$-损失）结合时，会产生一种潜在的结构失配，导致出现时间依赖的奇异加权，从而放大了梯度对近似误差的敏感性。基于此观察，我们将预测-损失对齐形式化为流匹配训练的必要条件。我们证明，将目标函数重新对齐至信号空间（$x$-损失）可消除奇异加权，产生一致有界的梯度，并使得在均匀时间步采样下无需依赖启发式调度即可实现鲁棒训练。最后，在确保对齐的基础上，我们研究了针对二元数据特有的设计选择，揭示了概率目标（如交叉熵）与几何损失（如均方误差）之间依赖于拓扑结构的差异。这些结果共同为二元（及相关离散）域上的鲁棒流匹配提供了理论基础与实践指导，确立了信号空间对齐作为鲁棒扩散学习的关键原则。