Heavy-Tailed Diffusion Models (2410.14171v2)

Abstract: Diffusion models achieve state-of-the-art generation quality across many applications, but their ability to capture rare or extreme events in heavy-tailed distributions remains unclear. In this work, we show that traditional diffusion and flow-matching models with standard Gaussian priors fail to capture heavy-tailed behavior. We address this by repurposing the diffusion framework for heavy-tail estimation using multivariate Student-t distributions. We develop a tailored perturbation kernel and derive the denoising posterior based on the conditional Student-t distribution for the backward process. Inspired by $\gamma$-divergence for heavy-tailed distributions, we derive a training objective for heavy-tailed denoisers. The resulting framework introduces controllable tail generation using only a single scalar hyperparameter, making it easily tunable for diverse real-world distributions. As specific instantiations of our framework, we introduce t-EDM and t-Flow, extensions of existing diffusion and flow models that employ a Student-t prior. Remarkably, our approach is readily compatible with standard Gaussian diffusion models and requires only minimal code changes. Empirically, we show that our t-EDM and t-Flow outperform standard diffusion models in heavy-tail estimation on high-resolution weather datasets in which generating rare and extreme events is crucial.

• The paper introduces a heavy-tailed diffusion approach using a multivariate Student-t distribution to better capture rare events.
• It develops a specialized denoising framework and extends models (t-EDM, t-Flow) to enhance tail estimation, validated by improved statistical metrics.
• The method integrates seamlessly with existing Gaussian diffusion models, offering practical benefits for forecasting extreme weather events.

Heavy-Tailed Diffusion Models: Enhancing Generative Modeling for Rare Events

The exploration of generative models capable of accurately capturing rare or extreme events is crucial in fields such as weather forecasting. Traditional diffusion models, while successful in generating high-quality samples, struggle with the accurate depiction of rare, heavy-tailed events due to their reliance on Gaussian priors. This paper proposes a refined approach using heavy-tailed distributions, specifically multivariate Student-t distributions, to enhance the diffusion process's ability to model such phenomena.

Methodology

The authors introduce modifications to conventional diffusion models to incorporate heavy-tailed characteristics. The key innovations include:

1. Student-t Distribution: The foundational Gaussian distribution used in traditional models is replaced by a multivariate Student-t distribution. This adjustment enables the model to allocate more density to the tails, thereby better capturing rare events.
2. Denoising Framework: A specialized denoising posterior is derived using the conditional Student-t distribution for the backward process. This is coupled with a training objective inspired by $\gamma$-divergence to optimize the model's ability to capture tail-heavy behavior.
3. Framework Compatibility: The proposed model, referred to as t-Diffusion, is designed to be compatible with existing Gaussian diffusion models, requiring minimal modifications to the codebase.
4. New Model Variants: The paper introduces t-EDM and t-Flow, extensions of existing diffusion and flow models incorporating the Student-t prior, aiming to enhance tail estimation capabilities.

Empirical Results

The empirical analysis conducted on high-resolution weather datasets demonstrates the efficacy of the t-EDM and t-Flow models in modeling heavy-tailed distributions. These models outperform standard diffusion models in capturing extreme weather events, as evidenced by:

• Improved statistical metrics such as the Kurtosis Ratio, Skewness Ratio, and Kolmogorov-Smirnov statistic, indicating better tail estimation.
• Enhanced qualitative performance in representing the distribution of rare occurrences.

Theoretical Implications

The work extends the theoretical framework of diffusion models by integrating robust statistical techniques, specifically $\gamma$-power divergences, to handle heavy-tailed data. This approach not only provides a robust estimation mechanism but also offers a controllable parameter to adjust tail estimation, thus providing flexibility in modeling diverse data distributions.

Practical Implications and Future Directions

Practically, the application of this framework to weather prediction exemplifies its potential to impact fields where extreme event modeling is critical. The method's compatibility with existing models ensures ease of integration and scalability. Future advancements may include learning the degree of freedom parameter for more dynamic tail adaptation and extending the framework to other domains requiring robust tail estimation.

This paper successfully addresses a significant limitation in current generative modeling techniques by enhancing the ability to capture rare and impactful events through a heavy-tailed diffusion framework. This contribution not only broadens the horizon for diffusion model applications but also sets the stage for future research in robust generative modeling methodologies.