Optimal transport with prior, introduced in Chen, Georgiou & Pavon (2014), is a generalization of the Benamou-Brenier formula that incorporates a prior velocity field v(x,t). Instead of minimizing the total kinetic energy of the transport velocity, it minimizes the deviation from the prior:
subject to the continuity equation ∂ρ̃/∂t + ∇·(ṽρ̃) = 0 with ρ̃(0) = ρ₀ and ρ̃(1) = ρ₁.
The standard Benamou-Brenier problem is recovered when v ≡ 0. The optimal velocity is ṽ*(x,t) = v(x,t) + ∇ψ(x,t), where ψ satisfies the Hamilton-Jacobi equation ∂ψ/∂t + v·∇ψ + ½||∇ψ||² = 0 with appropriate boundary conditions.
This formulation is the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution with drift v(x,t). It provides a natural framework for optimal transport when there is already a known dynamics (e.g., a physical prior like an Ornstein-Uhlenbeck process), and one seeks the closest transport plan to that dynamics. The cost function c(x,y) = inf_{x∈X_{xy}} ∫₀¹ ||ẋ - v(x,t)||² dt depends on v and is generally non-trivial to compute.
This concept connects strongly to the rectified flow framework and the bridge-flow duality in Connecting Brownian and Poisson Random Bridges with Rectified Flows: the random bridge drift can be viewed as the optimal stochastic perturbation of a prior transport, and the zero-volatility limit recovers exactly the OT-with-prior solution.
Key Details
- Generalizes Benamou-Brenier with a prior velocity field v(x,t)
- Optimal correction: ṽ* = v + ∇ψ (prior + gradient correction)
- Zero-noise limit of SB with Markovian prior: as ε→0, SB → OT with prior
- In the Gaussian case (A(t) linear, Gaussians marginals): reduces to matrix Riccati equations
- Standard OT recovered when v ≡ 0
- The transport cost c(x,y) becomes path-dependent when v ≢ 0