针对凸域生成建模中分布尾部过重的问题，提出新方法提升训练稳定性和样本质量。
Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains
- 使用正则化镜映射控制对偶分布尾部行为，确保矩有限
- 采用t分布先验匹配重尾目标，显著改善训练稳定性
- 理论保证速度场光滑性与收敛率，适合有约束生成任务
我们研究在凸域上基于流匹配与镜映射的生成建模，识别出两个根本挑战：其一，标准对数障碍镜映射导致对偶分布重尾，引发病态动力学；其二，与高斯先验耦合在匹配重尾目标时表现不佳。为解决此问题，我们提出基于正则化镜映射的镜流匹配方法，可调控对偶分布尾部特性并保证矩有限，并结合学生t分布先验以匹配重尾目标、稳定训练过程。我们提供了理论保证，包括速度场的空间Lipschitz连续性与时间正则性，以及使用ε-精确学习速度场时的Wasserstein收敛速率，和对偶空间的约束生成保证。实验表明，该方法在合成凸域模拟中优于基线，在真实世界约束生成任务中也达到具有竞争力的样本质量。
原文摘要 · Abstract (English)
We study generative modeling on convex domains using flow matching and mirror maps, and identify two fundamental challenges. First, standard log-barrier mirror maps induce heavy-tailed dual distributions, leading to ill-posed dynamics. Second, coupling with Gaussian priors performs poorly when matching heavy-tailed targets. To address these issues, we propose Mirror Flow Matching based on a \emph{regularized mirror map} that controls dual tail behavior and guarantees finite moments, together with coupling to a Student-$t$ prior that aligns with heavy-tailed targets and stabilizes training. We provide theoretical guarantees, including spatial Lipschitzness and temporal regularity of the velocity field, Wasserstein convergence rates for flow matching with Student-$t$ priors and primal-space guarantees for constrained generation, under $\varepsilon$-accurate learned velocity fields. Empirically, our method outperforms baselines in synthetic convex-domain simulations and achieves competitive sample quality on real-world constrained generative tasks.
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