Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains
This work addresses constrained generative modeling for domains like physics or finance, but appears incremental as it builds on existing flow matching and mirror map techniques.
The paper tackles generative modeling on convex domains by addressing challenges with heavy-tailed distributions in flow matching, proposing a regularized mirror map and Student-t prior coupling. The method shows theoretical guarantees and outperforms baselines in synthetic simulations while achieving competitive sample quality on real-world tasks.
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.