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

arXiv: https://arxiv.org/abs/2510.08929 · HTML: https://mlsanity.com/paper/2510.08929/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 30 of its 36 references that are in the map.
- First posted on arXiv 10 Oct 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 30)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Rethinking FID: Towards a Better Evaluation Metric for Image Generation](https://mlsanity.com/paper/2401.09603/) (2023)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024)
- [FlowPolicy: Enabling Fast and Robust 3D Flow-based Policy via Consistency Flow Matching for Robot Manipulation](https://mlsanity.com/paper/2412.04987/) (2024)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Heavy-Tailed Diffusion Models](https://mlsanity.com/paper/2410.14171/) (2024)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Evaluating the design space of diffusion-based generative models](https://mlsanity.com/paper/2406.12839/) (2024)
- [Mirror Descent Strikes Again: Optimal Stochastic Convex Optimization under Infinite Noise Variance](https://mlsanity.com/paper/2202.11632/) (2022)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023)
- [Reflected Flow Matching](https://mlsanity.com/paper/2405.16577/) (2024)
- [Preference Alignment with Flow Matching](https://mlsanity.com/paper/2405.19806/) (2024)
- [On the Convergence and Straightness of Rectified Flow](https://mlsanity.com/paper/2410.14949/) (2024)
- [Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling](https://mlsanity.com/paper/2502.09306/) (2025)
- [Neural Approximate Mirror Maps for Constrained Diffusion Models](https://mlsanity.com/paper/2406.12816/) (2024)
- [Constraint-Aware Flow Matching via Randomized Exploration](https://mlsanity.com/paper/2508.13316/) (2025)

## Built on by (most cited first, 8 of 8)

- [Discrete State Diffusion Models: A Sample Complexity Perspective](https://mlsanity.com/paper/2510.10854/) (2025)
- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training](https://mlsanity.com/paper/2605.13175/) (2026)
- [Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching](https://mlsanity.com/paper/2512.01286/) (2025)
- [Debiased Counterfactual Generation via Flow Matching from Observations](https://mlsanity.com/paper/2605.07665/) (2026)
- [PolyFlow: Safe and Efficient Polytope-Constrained Flow Matching with Constraint Embedding and Projection-free Update](https://mlsanity.com/paper/2606.13400/) (2026)
- [Fine-Tuning Generative Models for Extreme Events via CVaR-Penalized Wasserstein Gradient Flows](https://mlsanity.com/paper/2608.11544/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)

## Abstract (arXiv)

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 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.