OT displacement interpolation is the rectified flow fixed point

The optimal transport displacement interpolation satisfies the rectified flow straightness condition $S(Z)=0$ in any dimension $d\ge 1$, and is therefore the fixed point of the reflow procedure. The argument has four steps:

1. Individual paths are straight lines. For any strictly convex Lagrangian $L(v)$ in ${\mathbb{R}}^d$, Jensen’s inequality forces action-minimising curves to have constant velocity: $c({\gamma }_1-{\gamma }_0)=c({\int }_0^1{\dot{\gamma }}_{t}dt)\le {\int }_0^{1}c({\dot{\gamma }}_t)dt$, with equality iff ${\dot{\gamma }}_t$ is constant (Example 7.2, p. 128 of Villani 2009). So the OT displacement interpolation traces ${\gamma }_t^x=(1-t)x+t\nabla \phi (x)$.

2. The map is injective, so paths don’t cross. By Brenier’s theorem, $T=\nabla \phi$ for a convex $\phi$. The interpolation map $x\mapsto (1-t)x+t\nabla \phi (x)=\nabla ((1-t)|x{|}^2/2+t\phi (x))$ is the gradient of a strictly convex function, hence injective for all $t\in [0,1]$. This is confirmed by the non-crossing property: Theorem 7.30(iv), p. 151 of Villani states $[{\gamma }_t={\tilde{\gamma }}_t]⟹[\gamma =\tilde{\gamma }]$ a.s.

3. The conditional expectation is trivial. With the OT coupling $(X_0,X_1)\sim \hat{\pi }$ where $X_1=\nabla \phi (X_0)$, the linear interpolant $X_t=(1-t)X_0+t\nabla \phi (X_0)$ is a deterministic function of $X_0$. By injectivity (Step 2), $X_0$ is uniquely determined from $X_t$, so the rectified flow velocity $v^{\ast }(z,t)=\mathbb{E}[X_1-X_0\mid X_t=z]$ is deterministic — no averaging, no noise. The straightness measure $S(Z)=\int \mathbb{E}[\Vert Z_1-Z_0-{\dot{Z}}_t{\Vert }^2]dt=0$.

4. Reflow converges to this fixed point. The reflow procedure provably reduces $S(Z)$ at rate $O(1/K)$ (Liu et al. 2022, Theorem 3.5). Since $S(Z)=0$ iff paths are straight and non-crossing (i.e., the coupling is deterministic and OT-like), the OT displacement interpolation is the unique fixed point with $S=0$.

The chain of limits:

$\text{Schro¨dinger Bridge}\stackrel{\sigma \to 0}{\to }\text{OT (straight, non-crossing)}=\text{Rectified Flow fixed point}$

This connects three frameworks: the Γ-convergence of Schrödinger to Monge-Kantorovich (Léonard 2014), the Benamou-Brenier formula (OT as kinetic energy minimisation), and rectified flow (learned velocity field). The OT solution sits at the intersection as the deterministic, straight-line, minimum-energy transport.

Key Details

• Holds for any $d\ge 1$: the argument uses only Jensen’s inequality (Step 1) and convexity of $\phi$ (Step 2), both dimension-free
• Requires ${\mu }_0$ absolutely continuous w.r.t. Lebesgue (for Brenier’s theorem) — otherwise the OT map may not exist
• The OT map may lack smoothness in $d>1$ (Monge-Ampère singularities), but paths are still individually straight
• On Riemannian manifolds: paths are geodesics (not straight lines), so “straightness” generalises to “constant-speed geodesic”
• The displacement interpolation is a constant-speed geodesic in $(P_p,W_p)$: $W_p({\mu }_s,{\mu }_t)=|t-s|W_p({\mu }_0,{\mu }_1)$ (Corollary 7.22, p. 139)

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