Geodesics and Displacement Interpolation

Displacement interpolation is the mathematical foundation unifying diffusion models, flow matching, and optimal transport. Stable Diffusion literally travels a geodesic in measure space.

• **Rectified Flow (Liu 2022):** optimal flow = straight paths = displacement interpolation. Stable Diffusion 3 and Flux use this for 10x speedup.
• **Color transfer (Reinhard 2001):** displacement interpolation for transferring color palette between images. Used in film post-production.
• **Shape morphing:** geodesic between two 3D shapes = smooth deformation. Applied in medical imaging (MRI brain atlases).

McCann Interpolation: Geodesics in Wasserstein Space

**Diffusion models (Stable Diffusion, 2022) implement displacement interpolation in probability measure space , a 14-step path from noise to a 512×512 image.** A geodesic in (𝒫₂, W₂) is not a mixture of distributions but a simultaneous straight-line motion of every particle of mass.

Displacement interpolation differs from a linear mixture (1−t)μ₀ + tμ₁: the mixture creates a bimodal distribution, while the geodesic smoothly translates mass. Diffusion models travel the geodesic from N(0,I) to p_data.

Brenier's Theorem: Existence of the Optimal Transport Map

Monge sought a deterministic map T: X→Y transporting μ to ν at minimum cost. **Brenier's theorem (1991)** establishes existence and uniqueness for the quadratic cost: T = ∇φ for some convex function φ.

Brenier's theorem requires absolute continuity of μ (no atoms). In the discrete case the optimal map may be non-unique , use the Kantorovich plan instead.

Benamou-Brenier Formula: Dynamic Formulation of W₂

The static W₂ formulation searches over plans γ. The **Benamou-Brenier formula (2000)** reformulates it as a fluid dynamics problem: find a velocity field v(x,t) transporting ρ₀ to ρ₁ with minimum kinetic energy.

The Benamou-Brenier formula is the basis for numerical W₂ solvers via convex saddle-point optimization. Flow matching in diffusion models directly parametrizes the optimal velocity field v*.

Key Ideas

• **Displacement interpolation:** μ_t = ((1−t)id + tT*)#μ₀. Each particle moves in a straight line to its image under the optimal map.
• **Brenier's theorem:** optimal map T* = ∇φ (gradient of a convex function). In 1D: T = F_ν⁻¹ ∘ F_μ.
• **W₂² via quantiles (1D):** W₂²(μ,ν) = ∫₀¹ |F_μ⁻¹(q) − F_ν⁻¹(q)|² dq.
• **Benamou-Brenier:** W₂² = min kinetic energy of flow. Optimal velocity v* = T(x)−x (constant along the geodesic).

Связанные уроки

• ot-18 — Uses the W₂ metric from the previous lesson
• ot-14-gradient-flows — Geodesics underlie Wasserstein gradient flows