The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space
The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, no...
| Authors | Yian Yao, Weiwei Zhang |
|---|---|
| Primary category | cs.AI |
| Published | 23 June 2026 |
| Last updated at source | 23 June 2026 |
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