Unifying Diffusion and Flow Matching Models with Wasserstein Geometry.

This research paper provides a unified geometric understanding of two prominent generative AI techniques: diffusion models and flow matching. It posits that both methods can be understood through the lens of the quadratic Wasserstein space, a natural geometry for probability measures. The paper explains that diffusion models, which underpin many modern image and data generation tasks, essentially perform a gradient descent along the free energy landscape within this Wasserstein manifold, with each denoising step analogous to a JKO scheme. Conversely, optimal-transport flow matching models are shown to follow the geodesics—the shortest paths—within the same Wasserstein manifold. This means that while diffusion models solve an initial-value problem by descending a gradient, flow matching solves a boundary-value problem by tracing a direct path between two points. This unified perspective clarifies their relationship, demonstrating that they achieve similar generative outcomes through fundamentally different, yet geometrically related, variational principles.