Tochastic Interpolants: A Unifying Framework For Flows And Diffusions.

Scott Simon

2 min read

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11-01-2025

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The field of generative modeling has witnessed a surge in popularity, driven largely by advancements in diffusion models and normalizing flows. While seemingly disparate, these powerful techniques share underlying principles that can be elegantly unified under the umbrella of stochastic interpolants. This framework offers a fresh perspective, revealing the common threads connecting these seemingly different approaches and paving the way for future innovations.

Understanding the Core Concept

At its heart, a stochastic interpolant describes a continuous-time stochastic process that bridges a prior distribution (often a simple Gaussian) and a target data distribution. This interpolation is achieved through a carefully designed stochastic differential equation (SDE) or its discrete-time counterpart. Both diffusion models and normalizing flows can be viewed as specific instances of this broader framework.

Diffusion Models: Reverse-Time Diffusion

Diffusion models generate samples by reversing a carefully engineered diffusion process. This process gradually adds noise to data until it reaches a simple, tractable distribution. The reverse process, trained via score matching or similar techniques, learns to remove noise step-by-step, ultimately producing samples from the target distribution. Within the stochastic interpolant framework, this reverse diffusion process constitutes a specific interpolation path from the noisy prior to the complex data distribution.

Normalizing Flows: Deterministic Transformations

Normalizing flows, on the other hand, employ a series of invertible transformations to map a simple prior distribution to the complex target distribution. These transformations, typically implemented as neural networks, are deterministic, unlike the stochastic nature of diffusion processes. However, we can interpret these deterministic transformations as a special case of stochastic interpolants where the stochasticity is essentially zero along the interpolation path. This provides a unifying perspective, highlighting the fundamental similarity in their goal: transforming a simple distribution into a complex one.

The Advantages of a Unified Framework

The unification offered by the stochastic interpolant framework provides several key advantages:

• Enhanced Understanding: It provides a clearer, more intuitive understanding of the relationship between diffusion models and normalizing flows. This facilitates a deeper appreciation of their strengths and weaknesses.
• Facilitated Innovation: The framework allows researchers to draw inspiration from both approaches, potentially leading to hybrid models that combine the best features of each. For example, incorporating stochasticity into normalizing flows or exploring deterministic components within diffusion models.
• Improved Theoretical Analysis: By placing both methodologies under a common theoretical umbrella, it simplifies the analysis and comparison of their properties, such as sample quality and training stability.

Future Directions and Open Questions

While the stochastic interpolant framework provides a powerful unifying lens, several intriguing open questions remain:

• Optimal Interpolation Paths: Determining the optimal interpolation path – the specific SDE or sequence of transformations – remains a significant challenge. Research into designing more efficient and effective paths is critical for improving the performance of generative models.
• Hybrid Models: Exploring the development of hybrid models that seamlessly integrate aspects of both diffusion models and normalizing flows presents a fertile ground for future research.
• Theoretical Guarantees: Further theoretical investigation is needed to establish rigorous guarantees regarding the convergence and stability of stochastic interpolants under various conditions.

The stochastic interpolant framework promises to be a valuable tool for advancing the field of generative modeling. By offering a unifying perspective, it unlocks new possibilities for research and innovation, ultimately leading to the development of even more powerful and versatile generative models.