Heavy-Tailed Flow Matching via Random Clocks

2026-07-15

AI Summary

Standard diffusion and flow matching models rely on Gaussian source distributions, which struggle to capture rare yet critical events in heavy-tailed data such as imbalanced images, financial returns, and extreme weather. This work proposes a Heavy-Tailed Flow Matching (HTFM) framework that introduces a stochastic clock mechanism, modeling the heavy-tailed source as a clock-conditioned Gaussian mixture: conditioned on a clock path, the flow field retains a Gaussian structure, while marginalization over the clock yields flexible heavy-tailed marginals—including Gaussian, α-stable, and Student-t distributions. The method employs truncated log-signature encodings of clock paths to render the velocity field adaptive to the conditioning space while remaining computationally efficient. Experiments demonstrate that HTFM significantly outperforms existing approaches on 2D α-stable mixtures, CIFAR10-LT, and HRRR weather data, achieving superior mode coverage, sample quality, and recovery of tail statistics, all while maintaining low NFE sampling efficiency.

Abstract

Heavy-tailed data arise in many domains where rare events carry disproportionate importance, such as imbalanced image datasets, financial returns, and weather extremes. Standard diffusion and flow-matching models typically begin from Gaussian noise or Gaussian source distributions, which yield tractable training targets but provide a poor inductive match for heavy-tailed data. We propose Heavy-Tailed Flow Matching via Random Clocks (HTFM), a framework that portrays heavy-tailed sources as mixtures of clock-conditioned Gaussian sources. Conditioning on a given clock path, the source distribution and flow are Gaussian; marginalizing over the clock gives a Gaussian scale mixture covering Gaussian, $α$-stable, and Student-t families. To make the clock-conditioned vector field practical, we encode the path-valued clock using truncated logsignature features, allowing the velocity field to adapt to the realized conditional space with negligible overhead. Empirically, on 2D imbalanced $α$-stable mixtures, CIFAR10-LT, and HRRR weather fields, HTFM improves mode coverage, sample quality, and tail-statistic recovery over Gaussian flow matching and competitive heavy-tailed baselines, while retaining the low-NFE sampling advantage of flow matching. Moreover, the random-clock formulation further provides a practical tail-control interface: by varying only the clock law or tail parameter, the same architecture can calibrate the ``heaviness'' of generated tails across different distribution families.

Problem

• heavy-tailed data
• flow matching
• Gaussian assumption
• tail modeling
• imbalanced data

Innovation

• Heavy-tailed Flow Matching
• Random Clocks
• Gaussian Scale Mixture
• Logsignature Features
• Tail Control