Random Clocks for Heavy-Tailed Flow Matching

Introduction

Many real-world datasets are not well described by light-tailed Gaussian assumptions. In long-tailed image collections, financial returns, and severe weather fields, rare samples may be sparse but disproportionately important. Standard diffusion and flow-matching models usually start from Gaussian noise or Gaussian source distributions because they are mathematically convenient, yet this choice can be a weak inductive match for heavy-tailed data.

The arXiv paper “Heavy-Tailed Flow Matching via Random Clocks” proposes a framework called HTFM. Its central idea is to keep the tractability of Gaussian conditional flows while recovering heavy-tailed behavior after marginalizing over a random clock.

Key ideas

• A different view of the source distribution: Instead of treating the source as a fixed Gaussian, HTFM represents heavy-tailed sources as mixtures of Gaussian sources conditioned on clock paths.
• Random clocks as a bridge: For a given clock path, both the source distribution and the flow remain Gaussian and easier to handle. Once the clock is integrated out, the resulting distribution becomes a Gaussian scale mixture that can cover Gaussian, α-stable, and Student-t families.
• Practical clock conditioning: A clock path is not a simple scalar feature. The paper uses truncated logsignature features to encode the path, giving the velocity field information about the realized conditional space with reportedly negligible overhead.
• Maintaining efficient sampling: Flow matching is often valued for low-NFE sampling. The authors state that HTFM preserves this advantage while improving behavior in the tails.
• Evaluation on long-tail and extreme data: The reported experiments include two-dimensional imbalanced α-stable mixtures, CIFAR10-LT, and HRRR weather fields. Across these settings, HTFM is said to improve mode coverage, sample quality, and recovery of tail statistics compared with Gaussian flow matching and competitive heavy-tailed baselines.

Why it matters

The contribution is not merely swapping one noise distribution for another. HTFM offers a structured way to introduce heavy tails into the generative process while keeping a conditionally Gaussian formulation that is easier to train and reason about. This is a useful design pattern for domains where extremes matter as much as averages.

The paper also highlights a potential control mechanism: by modifying only the clock law or tail parameter, the same architecture can calibrate how heavy the generated tails are across different distribution families. That could be valuable for stress testing, rare-event simulation, weather modeling, and generating underrepresented visual categories.

As with any new arXiv result, the full paper is needed to judge implementation details, metric choices, baseline fairness, and robustness in larger deployment scenarios. Still, HTFM points to an important direction for generative modeling: if the data-generating process is heavy-tailed, the source distribution should not always be forced to begin in a Gaussian world.

Source: arXiv