通过熵约束提升生成模型路径的语义多样性，避免模式崩溃。
Entropy-Controlled Flow Matching
- 在连续性方程路径上施加全局熵率约束，控制生成轨迹的信息几何。
- 理论证明可避免低熵瓶颈，实现模式覆盖与密度下界保证。
- 适合关注生成质量、模式多样性与理论稳定性的研究者。
现代视觉生成模型通过时间索引测度将基础分布传输至数据分布，实现方式为确定性流（ODE）或随机扩散（SDE）。尽管性能优异，标准流匹配目标不直接控制轨迹的信息几何，可能导致低熵瓶颈，临时耗尽语义模式。本文提出熵控流匹配（ECFM）：一种在连续性方程路径上的约束变分原理，强制熵率变化满足 d/dt H(μ_t) ≥ -λ。ECFM 在 Wasserstein 空间中为凸优化，具有 KKT/Pontryagin 系统结构，等价于带有显式熵乘子的 Schrödinger 桥。在纯传输情形下，ECFM 恢复熵正则化最优传输测地线，且当 λ → 0 时 Gamma 收敛至经典最优传输。我们进一步获得模式覆盖率和密度下界证书式保证，并构造了无约束流匹配的近似最优坍塌反例。
原文摘要 · Abstract (English)
Modern vision generators transport a base distribution to data through time-indexed measures, implemented as deterministic flows (ODEs) or stochastic diffusions (SDEs). Despite strong empirical performance, standard flow-matching objectives do not directly control the information geometry of the trajectory, allowing low-entropy bottlenecks that can transiently deplete semantic modes. We propose Entropy-Controlled Flow Matching (ECFM): a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t) >= -lambda. ECFM is a convex optimization in Wasserstein space with a KKT/Pontryagin system, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier. In the pure transport regime, ECFM recovers entropic OT geodesics and Gamma-converges to classical OT as lambda -> 0. We further obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz stability, and construct near-optimal collapse counterexamples for unconstrained flow matching.
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