Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains

2025-10-09

AI Summary

This work addresses generative modeling on convex domains via flow matching and mirror mapping, tackling two key challenges: (1) standard logarithmic barrier mirror maps induce heavy-tailed dual distributions, leading to ill-posed dynamics; and (2) Gaussian priors fail to align with heavy-tailed target distributions, causing training instability. To resolve these, we propose a novel framework coupling a regularized mirror map—with controlled tail behavior and guaranteed finite moments in the dual space—and a Student-*t* prior, explicitly designed to accommodate heavy tails. We establish, for the first time, Wasserstein convergence rate guarantees and rigorous theoretical foundations in the primal space for constrained generative modeling. Experiments demonstrate significant improvements over baselines on synthetic convex-domain data and produce high-quality samples on real-world constrained generation tasks, validating both theoretical soundness and practical efficacy.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

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.

Problem

• Addresses heavy-tailed dual distributions from log-barrier mirror maps
• Solves poor Gaussian prior coupling with heavy-tailed targets
• Enables constrained generative modeling on convex domains

Innovation

• Regularized mirror map controls dual tail behavior
• Student-t prior aligns with heavy-tailed targets
• Provides theoretical guarantees for constrained generation