The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space

• URL: http://arxiv.org/abs/2606.24157v1
• Date: Tue, 23 Jun 2026 05:25:24 GMT
• Title: The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space
• Abstract summary: It is shown that the gradient flow of the free energy F(rho) = KL(rho ) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme.<n>Its geodesics are precisely the optimal-transport paths that Flow Matching learns.
• Score: 1.636137854123538
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.

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